hey everybody Charles from GMAT ninja tutoring here today we're going to talk about combination and permutation questions on the GMAT and executive assessment sometimes known collectively as combinatorics questions this is one of those things people freak out all the time it's one of the Boogey men of the GMAT oh I'm terrible at it I hear it all the time it's one of the biggest things holding me back one of the biggest things I need to practice guess what guys these are actually really rare on the GMAT combinations permutations together maybe 2 or 3% of your
Quant questions at most the occasional Cameo on Data Insights if you're taking the GMAT fairly rare not a huge thing if you're going for a super Elite Quant score by all means you want to be really good at these even that 2 or 3% can really matter if your goals are pretty modest if you're looking for a 75 or an 80 80 is a fantastic score on the GMAT odds are pretty good that you're going to be in really good shape if you can Master the very very Basics defend yourself against missing easy questions so
first three questions we're going to go through today first two or three questions fairly fundamental I'll be really clear about kind of labeling when you're going to be in good shape if your goals are fairly modest when you might want to turn off the video for people going for that Elite score we're going to go through a total of seven questions today the last few are going to get pretty tough going to throw some little logical tweaks in there that are very much in the style of the GMAT have a little fun with you on
those give you a taste for the kinds of things they can do in those rare really hard combination permutation questions now usually we do a little bit of data sufficiency in these videos which obviously appears on the data Insight section of the G mat uh today we're not going to just because combination permutation questions and data sufficiency are exceedingly rare they could show up but there's just not that many of them out there we're going to focus on problem solving today now again this is part of our full free GMAT course we have up on
our YouTube channel if you're enjoying it please hit the like button please subscribe if you're really enjoying it and you're trying to figure out what's going on in your head want some one-on-one attention hit us up at GMAT ninja.com we've been doing 10one tutoring for more than 20 years now and if we're saving you money with this full free course consider noted some of our fa Char Charities and you can find a link to those in the video description all right with that let's jump into the first question go and take a couple minutes on
this one e e e all right this one's about a straightforward as a permutation question is going to get on the GM executive assessment nice simple setup some dude named Ralph visiting Bania via Bania Airlines going to to four destinations out of the airlines 10 now biggest thing most important thing by far on the GMAT or EA with commentator questions notice whether order matters or order doesn't matter they're not going to tryy to trick you with this they're not going to get sneaky with it very very clear pretty much every time when you see an
official question Ralph will visit the locations in the order in which they are chosen that's actually a line that we borrowed from an actual GMAT question the rest of the setup is ours that particular line is something that actually appears in a real jmat question why are we emphasizing that easiest way to make a mistake here is to apply the wrong formula or to think incorrectly about what's going on some people memorize formulas I'll talk more about that on the next question here just be intuitive picture what's going on Ralph here is going to go
to four places so he's got destination number one destination number two destination number three and destination number four we care about the order how many options does Ralph have in his first destination well 10 different destinations go to 10 different places how many are left for the next one you go to nine different places how many for the next eight remaining and then seven cuz he's visiting different destinations not repeating them so what are you doing here that's it you just multiply these 10 * 9 * 8 * 7 even if you don't want to
do that math it's got to be this big boy right here don't even need to do the arithmetic clearly has to be the right answer and that is it nice and straightforward if you like the jargon this is called a permutation if you don't like the jargon please don't worry about it some people find it easier to separate permutation combination if they hear those words they're never going to actually get used on the GMAT if that jargon kind of makes your head spin don't worry about it do you notice whether order matters or not that's
it skill number one if order does matter so visiting cities a b c and d is a different itinerary than visiting cities DC B and A in that order then order matters just use slots be nice and intuitive that's going to cover you on most permutation questions that you really need to get right on this test all right basic permutation that's that one now we're going to give you a slightly different variation on this question go a and take a minute or two e e e all all right setup here looks an awful lot like
question number one in that one we had our buddy Ralph visiting four destinations out of Bania Airlines 10 but notice the language we cared about the order it said that really explicitly in the question so when that's the case you're going to use slots again picture what's going on choose one destination how many options there were 10 how many for the next one there were nine and so on same deal here combinations permutations it has so much logic in it typically and job number one is picture the setup that's what's going to unlock everything for
you even as they get crazy crazy hard down the road what's going on here okay somebody's going to two of the six domestic destinations two of the four International destinations they're going to separate those out right imagine that you're planning a vacation you're going to say all right I've got six domestic options I'm going to choose two of those right now as soon as you hear that word choose that's your hint this is combinations not permutations order doesn't matter at all we're not going to use slots instead when order doesn't matter we're using a nice
little formula if you've never seen it before I'm going to explain it quickly okay so dealing first with these this business of choosing two domestic destinations out of six what is n n is the number in kind of the large group in this case we're choosing among six different domestic destinations K is the number that you have in your subgroup so in this case we're choosing two out of those six destinations from there it really is just plug in chug 6 factorial over 4 factorial * 2 factorial this is going to be the number of
ways to choose when order doesn't matter choose two destinations out of that total of six domestic destinations now quick little thing here you could write out all these numbers that's not a habit I want you to get into I'm going to do it here just so you can see what's going on 6 * 5 * 4 * 3 * 2 * 1 I'm going to give myself a hand cramp because I love you not even going to charge you for it 4 * 3 * 2 * 1 * 2 * 1 again notice that you
could have canceled out the 4321 from the start as you practice these problems if you're new to them that's going to hopefully become second nature I want you to think about it consciously as you're just starting out 6 factorial divided 4 factorial what I could do instead is just write out 6 * 5 on top from the very start canceling out everything from four downward on the bottom I've just got the two that's going to give me 15 that's the total number of ways to choose two destinations out of the six domestic options all right
what about the other half of the question International destinations I've got four to choose from I'm going to choose two of them so same idea four factorial up here 2 factorial 2 factorial and again if I'm being really enterprising a little bit more efficient um I don't have to worry about anything from two downward up here 4 * 3 I've still got that two hanging out down there that's going to be six okay so 15 different pairs of two domestic destination I could choose from six six different pairs of two International destinations what do I
do with those numbers really common mistake I see here people will try to add them up but think about what's going on here so let's suppose that these domestic destinations have I don't know let's let's say that they're lettered from a to F so we could have a and b we could have B and C we could have a and d and so on so 15 different pairs if I were to write them all out pain in the butt to do so but just picture okay I've got 15 different pairs for each of those pairs
I've got six different pairs of international destinations so if those have let's say numbers on them so I could have 1 two or 1 three or 2 three and so on so for each pair of domestic destinations I have six different pairs of international destinations so what am I doing here am I adding or I multiplying soon as you hear that phrase for each of these you're multiplying 16 sorry 15 * 6 that's going to give you 90 and that's our answer okay guys if your goals are pretty modest if you're taking the executive assessment
applying to an executive NBA program you just need a 10 11 12 something like that on Quant if you're taking the GMAT you just need something like a 75 or even an 80 to be happy on Quant you know what you could probably stop here if you can handle this here's when order matters here's when it doesn't matter you keep those two separate you read careful you can picture the setup you're probably okay here you can stop watching the video If you really want to um next questions maybe a little Notch up after that it's
going to get Haier but the joke of the GMAT is these aren't that common 2 to 3% of the question are going to be combinations and permutations you don't need to be all that great to get a really at this stuff specifically to get a really solid score so if you can handle these two questions and you're feeling really good about it you're already in really really good shape to get a strong score all right going to go and kick this up a notch for those of you want to stick around good and take a
minute or two on this one e e e all right if you need a little more time on this one feel free to hit the pause button rewind as needed okay so again job number one on these questions is this a situation where order matters or it doesn't matter they're not getting tricky here they're always going to be really clear combinations of four items can be selected we're in the world of combinations order does not matter we're going to use the formula we're not going to use the slots okay great now what makes this question
a little bit funky what's this business of at least one you're going to see that not infrequently at least you're going to see it sometimes on probability questions occasionally on combinations permutations again what is this all about with the GMAT there's not a whole lot more technical stuff here on combinations permutations it's that I kind of find remarkable in general about the GMAT they take basic basic stuff they layer up these questions that have more and more layers of logic and reasoning and they turn into a really hard puzzle there's usually not a lot technical
there this is one of those cases really it's just thinking through okay picture this situation and can you kind of find a way to int your way to what you need to do here if I walk in a restaurant there's six salads there's four desserts on the menu and I go okay I want to select something that has at least one dessert I want four items cuz I'm really hungry I'm going to select four items but there's got to be at least one dessert in there how many ways are there to do that okay well
what's the the compliment or the opposite of getting at least one dessert I can do one or two things when I walk into this restaurant I can get at least one dessert or I can get no desserts one of those things is going to make me happier than the other obviously so in this case I either get at least one dessert or I get no desserts right okay that's the key once you get that you basically unlock the question how many total combinations are there let's start there so total combinations all right 10 items I'm
going to choose four of them so 10 factorial over 6 factorial four factorial again little trick here I'm going to slice off everything from six downward I've got 10 * 9 * 8 * 7 over 4 * 3 * 2 again I'm not going to multiply all this out yet going to see what I can cancel out really nice here I've got the four and the two cancel the eight and I've got the three that can cancel the nine down to a three so three * 7 21 * 10 I've got 210 combinations not done
yet though that's total combinations regardless of whether they include a dessert I only want the ones that have at least one dessert could be four desserts could be three desserts could be two desserts could be one dessert I don't want to calculate all four of those things what I can do instead is say here's my total combinations I'm going to subtract out the combos that have no desserts so combos with no dessert the ones that make me sad okay so what's that that would just be four salads right how many ways can I choose four
salads from a group of six salads we're right back to using that formula this time six salads to choose from I'm going to choose four and we're almost done how many ways are there to get four salads no desserts 6 * 5 / it two and we've got 15 so total combinations regardless of how many desserts there are combinations with salad only no desserts 210 minus 15 195 that's it we're done now these questions we can start layering them up we can make them almost infinitely difficult and it's not that there's anything technical there very
very little that's technical or or things I can teach you where it's like hey here's a Nifty new formula for this other kind of combination or permutation question very very little that exists for this test instead it's just when they give you a little tweak I'm looking for at least one dessert in my four items can you kind of Inuit your way through it apply a little bit of logic figure out what to do nothing super technical now the rest of this video These are going to get tougher but it's the same Basics constantly I'm
just adding layers these questions are going to get harder and harder again if your goal is an 80 on GMAC Quant something like a 10 11 12 on the executive assessment Quant you probably don't need the rest of this video If you think you're going to enjoy it stick with us fun to have you here um but unless you're going for Elite score you really don't need to master the rest of this video all right with that said if you want to stick with us thank you for sticking around we'll try to have fun go
and Tackle this next question e e e all right now the party starts these are getting pretty tough again for many of you watching this if your goal is something like an 80 or below on GMAT Quant you don't need this we're having a good time these are kind of fun they're kind of interesting they're really really tough at times but again same basic same thing we did in the first two questions just with a little wrinkle here there those wrinkles tend to be logical if you can apply a little bit of intuition here and
kind of think about what's going on you don't need anything technical is there a formula for tack tackling this question yeah do you need it no if you can kind of think through what's going on you can unwind this even if you haven't explicitly learned a formula or any technique behind this all right in this situation again what jumps out at me right off the bat got letters o i n n n and K can be rearranged okay right there I know right off the bat we're talking about Arrangements order matters this is going to
be about slots not the formula for combinations we're going to be using slots order matters sixlet strings um word strings isn't necessarily something we use in this context in the real world very often but codes would be another way to say it if we rearrange these letters we get a different code so we know we're talking about slots here how many distinct sixlet strings can be created all right if you're playing with this my first thought here when I see something like this is all right sixlet strings how many options do I have um well
I've got six slots I've got six options for the first five for the second four for the thir 3 2 1 I could multiply all those together if I do that I'm going to get 720 cool sounds good hold on Here's Where The Logical wrinkle comes in is you've got the same letter repeated so if I have this for example o i NN n k it's three different ends but they all look the same so another way to think about it is how many different three-letter strings could be created with three ends only one right
we're not saying that it's 3 2 and one because it's the same letter we can't tell the difference between for example if we gave these numbers and said this is N1 N2 and N3 well then yeah sure these are three distinct characters so we're going to have three options for the first if we're just rearranging the ends three options for the first slot two and then one there'd be six ways to do it but if we don't have these little subscripts here they all look the same there's only one way to rearrange it even though
it feels like there should be six again if these were distinct letters there'd be six ways to rearrange these three but since there aren't there's only one that's where this whole question unlocks the idea here the concept here is when we do this in this question when we just say 6 * 5 * 4 * 3 * 2 * 1 we're treating it as if all six letters are completely distinct and look distinct where is the redundancy coming in it's coming in because those NS all look the same so by what factor are we over
counting is the question here so three NS they are identical only one way to rearrange those three ends but if they were distinct there' be six so what we're saying is we're over counting by a factor of six so if we're doing it this way we're getting six times as many strings or codes or arrangements as we should so pretty straightforward from here once you recognize that bit of logic you're overcounting by a factor of six all we have to do is take this divide it by six we're done that's it 720 divide it by
6 or you could just cancel the six up here if you prefer same deal 720 divided 6 that's going to give us 120 and our answer is D now again if you struggled here please don't feel bad combinations permutations total 2 or 3% of your exam on the GMAT of the Quant section that's about it we can keep creating these logical wrinkles until the cows come home we can drive ourselves nuts the GMAT does get creative with these but again they're rare and there's not a lot technical here if you can follow the logic you'll
be able to unlock some of these really hard ones even if you haven't done a ton of combinations permutation sometimes you can follow the logic and do really well don't get intimidated by them there's usually less going on than it seems okay three more questions we're going to keep ramping them up a little bit and you're going to see this concept one more time in a harder question in a minute or two but uh one more first go and take a couple minutes on this next one e e all right these are getting tough now
um no shame at all for struggling rewind pauses you need to have some fun with this one uh like I said on the last couple questions really they're just making little logical tweaks nothing technical here typical of the GMAT typical of even the hardest questions sometimes there's a couple different ways to do this I'm going to show you one way of going about this that's really rooted in the logic of the question if you have a different way of doing it that works out for you fantastic I can think of a few others we're just
going to focus on one right now okay five countries each country sending three people to a conference so you got 15 people total we're going to choose a three-person committee that does not include more than one representative of a different country so you C so there can only be one person no more than one person from each country three people on the committee five countries represented what do we do here um again notice the language committees chosen we're in the world where order does not matter here it's going to be using the formula combinations not
permutations not slots going to use the formula okay cool now how the heck do we use the formula here that's the tough part maybe your instinct is well I've got 15 people total I'm going to choose three of those so 3 factorial * 12 factorial not a bad start you could do something with that but notice that this really doesn't take into account the whole business of well we can't have multiple people people from the same country again you could take this you can get somewhere with this it's doable I've got a way that for
me at least is a little more intuitive and that's what it's all about as you do these if you can get to the intuition that works for you run with it so here what I'm going to do instead is say wait a minute no more than one person per country so I could think of this as I'm really choosing three countries out of the five to be represented so step one again my way of doing it you could find a different one that's just as quick maybe a little bit slower lots of ways to do
it typical GMAC give you lots of solution paths do your best to pick the one that makes the most sense to you that you can execute on quickly for me I'm thinking hey I'm going to choose and there's that word again three countries out of the five easy enough 5 factorial 3 factorial 2 factorial 5 * 4 over 2 we've got 10 so that's 10 different trios of countries so there's 10 different ways to choose three countries out of these five great now let's picture this what are we doing with each with each of those
so I've got 10 ways to choose trios of countries so let's say we've chosen the countries I'm going to draw silly little circles here and say that this is country a and we've got person one two and three in there this is country b b for Bania very beautiful this time of year I hear the Rocky Mountain oysters are great terrible joke worst joke of the day ignore that uh so we've got three countries here each of them has three people so what do we need to do this number to to get to our answer
here well we've got three different people we can choose from the first country first of the three countries that we've chosen so three choices there so I'm going to take that 10 10 ways to choose three countries from five within the first country doesn't matter which one it is within that first country whatever it may be three different options for the individual people three different options here for the second country three different options for the Third Country multiply all that together we get 270 and and that's answer Choice B and look guys if your head's
spinning no shame in it I'll say it over and over again 2 to 3% of your questions are combinations permutations on the GMAT on the EA that's about it this is tough first time I saw it I saw something very close to this on an actual exam once years and years ago uh it took me a little while took me maybe four minutes to get through it first time I saw it no shame in it at all this is really good exercise for you to kind of grapple with the logic here but you're not going
to see a lot of stuff that's difficult unless you're way up there on the score scale and even if that's the case are you going to see this exact variation not necessarily but let's see if you can get better and better at applying logic thinking through the scenario finding some intuition that unlocks it all right two more questions both of them are going to be pretty tough go and take a couple of minutes on this next one e e e all right friends you're going to love this one it's really hard this is a really
nasty question could we find something harder yeah probably but it doesn't get much harder than this I know I'm a broken record on this but if you struggled here if you want to take some more time yourself out enjoy it see if you can get to the logic underneath it but there's a lot of layers really tough question you you it is very unlikely you're going to see something tougher on the GMAT and even if you do it's not going to Ru ruin your chance at a great score well into the 80s this is nasty
all right um again as always language should jump out at you rearranged strings again not a very common word that you see in day-to-day life don't worry about it too much they're always going to be pretty clear about this sort of thing they're not out to trick you order clearly matters here the Temptation we got the five letters we're going to turn into five letter string again the Temptation kind of like on the question we did that was like this a minute ago with uh I think it was the word oink with some extra NS
in it my son my three-year-old calls me Oinky what does that tell you about me uh so we could just say all right well there's five options here four options here three here two here one here but again that ignores the redundancy the ks look identical the O's look identical so this part of it's going to be very much like the one we did before so there are 5 * 4 * 3 * 2 total ways to arrange these letters if they were distinct but because the ks look alike two ways so there's a factor
of two there where you're going to have redundancy so we can divide by two so we got the five factorial here we divide it by two or two factorial same thing to kind of take out the redundancy of the K's same deal with the O's right so that's another factor of two where we're going to be over counting by a factor of two if we don't take that two out here so * 2 * 2 in the denominator so that's going to give us 5 * 3 * 2 if we do a little cancelling and
that's 30 different ways to make a felet string with these letters now of course we haven't dealt with uh this nasty little thing how many strings can be created in which the two Ks are not adjacent to each other oh fun times what do you do with that look sometimes on these combinatorics questions sometimes it's kind of the kind of dumb guy way that helps you a lot just try to Picture This try to do something with it my first thought on something like this again many ways to do these questions very typical GMAT some
of you are going to find an even more elegant solution than this this is what I find the most intuitive so I'm going to show you that way of doing it so the K's so we've got 30 total ways now if I can figure out the number of ways in which we can do these codes where the Ks are adjacent and subtract that I'm going to get my answer and be a happy guy so what does that look like well I could have KK at the beginning and then I've got three more I could have
a slot here KK here two more slots and so on right KK I'm running out of space KK here fantastic these all are going to end up kind of being almost identical okay so how many different strings can I get with 2ks at the beginning well the Temptation here again is to say 3 2 1 three options here two options here one option here six ways but again we got to think about that redundancy thing so there's going to two's out of these last three letters so we're going to be over counting doing it this
way by a factor of two so if the 2ks are at the beginning how many different strings could we get just three 3 * 2 divided by that two to to cancel out the over counting of the O's so we're going to get three different ways here with the case in the beginning this is going to be identical it's not 3 21 it's going to be 3 2 1 iD two to take out the redundancy with the O's so we've got four spots we could stick the Double K and for each of those four spots
we're going to have three different ways to arrange the rest of it so that gives us three scenarios with a k at the beginning three scenarios with a K's after a different letter there three here three here that's going to give us a total of 12 that's 12 different strings we could create where the Ks are adjacent I've got 30 total regardless of whether the K's are adjacent 30 minus 12 that's going to give me 18 and there we go again I know I'm broken record here no shame at all if you struggled lots of
layers to this one but notice really nothing technical there's nothing lurking there where you need some more formulas that's really not how the test Works we're just going to layer stuff up make it harder and harder all right last question of the day this one's about as nasty as it gets uh full disclosure when we created this question um I can't remember if I wrote it first or if Harry one of our other teers wrote it first um but we went back and forth about six times on it uh I think we missed it four
times total between the two of us before we figured out what we were doing wrong and adjusted the question it's tough so enjoy it do your best with it don't be embarrassed if you struggle go and take a minute e e e hey everybody sorry to interrupt your Quant viewing experience with that weird big noosed Tutor Guy chances are if you've made it this far in the video you're pretty serious about this whole gmet thing and if that's the case got two things I want you to be aware of first of all if at any
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than later get your head around the process start brainstorming a little bit I promise it's going to make your life much much easier down the road all right that's it if you've enjoyed these videos like subscribe you know the drill all right back to the Quant all right I'm resisting the urge to just laugh maniacally at you uh this is a really tough question if you sealed through this you've got an answer you're feeling confident awesome you deserve a cookie this is really really really nasty kind of the end of the rainbow you're just not
going to see anything that's a whole lot more difficult than this um so once again no shame at all if you struggled let's hop in there again plenty of ways to do it just like a lot of questions here we got four digigit positive integers they're greater than 4,000 how many of those numbers have three digits that are equal to each other and one that's different from the other three again it's just applying logic no extra formulas here no extra magical techniques here it's just thinking through how do you do this how do you make
sense of it now for me personally what I found the easiest here is to kind of take a number as an example and just use that to kind of think through what's going on so kind of first step for me again plenty of ways to do this if you found something that's more elegant and works for you awesome that's great for me I thought okay one possibility here so I've got four numbers four digits in this number let me just just so I have an example in my head to help me get my head around
it let's suppose the first digits a seven let's just say so between 7,000 and 7,999 fine okay so one way to fit this criteria one digit that's different from the other three is for all three of these to be not seven well how many ways are there to do that well you got the digits 0o through n that's actually 10 digits can't be a seven so there's nine digits that are not sevens so we've got nine different ways here to start with a seven and then have the other three match without being a seven okay
so nine ways there if we start with a seven thought number one I'm going to park that on the side for a second part number two well what if we've got the seven is the thing that repeats so suppose here we've got seven in the beginning and again just as as a way to get my head around it so what if this everything's a seven three sevens and what if this last one is the only one that is not a seven well how many ways are there to do that once again just like on the
last one we've got nine different digits that are not sevens so there's nine ways here as well okay but there's more we could stick that not seven in any of these three locations so there's n ways with the not seven here there's nine more ways here if all the others are sevens there's going to be nine different four-digit numbers that have three sevens and something other than a seven in that third spot same idea here there's going to be nine ways here so what's our total so far so we've got nine ways to have these
three match and not be a seven we've got nine ways for this to match to to get three of a kind and this is the non seven nine ways here nine ways here you add all that up step number three we get 36 total ways yeah but that's only if the first digit is a seven we've got numbers that are 4,000 and up so how many different starting numbers could we have we could have four 5 6 78 n that's six different starting numbers not just seven so four five6 789 go and count them on
your fingers no shame in that if that's what stops you're making a careless air knock yourself out 36 different ways for each starting number we've got six different starting numbers number 6 * 36 216 I'm happy great the answer is C hold on wait wait wait wait welcome to the GMAT again they're not out to get you they're not out to trick you you want those thoughts out of your head sometimes this is just a consequence of the language they didn't say 4,000 or greater they said greater than 4,000 the way we just did this
this would include the number 4,000 four in the first spot three zeros in the next three spots so we've overcounted by one one because the number has to be greater than 4,000 we got to subtract one from this and that's going to give us 216 all right congratulations on making to the end of the video if you got this one right um awesome work really really tough my point here we can throw wrinkles in here till the cows come home we can find different ways to kind of tweak the logic there really isn't a lot
of new stuff there these can feel super hard but things this tough are exceedingly rare you could see one or two of them if you're really a high flyer you might take the test a bunch of times never see anything even close to this so don't waste too much time on it if you if you really enjoy this or if you're going for a super high score knock yourself out keep working on these kinds of problems they're fun for you but don't go memorizing steps it's just applying logic to those very very very Basics all
right that wraps this up for today thank you everybody for watching if you enjoyed the video hit like hit subscribe uh if you need more help want some oneon-one help hit us up Gat ninja.com thank you so much have a great day everybody for
