GMAT Combinatorics Questions: Permutations vs. Combinations (plus Shorthand!)
As noted above, this is simply a clipped factorial.
Let’s take the case from above and modify it slightly.
Three paintings, from a group of seven total paintings, are chosen to be hung on a wall. In how many ways can these three paintings be arranged?
So: seven paintings to choose from, but need to figure out how many different ways (orders) we can hang the three paintings.
All we have to do is ask the three Core Questions, of course.
How Many Spaces?
There are three spaces to hang the paintings. That makes… three.
_ _ _
How Many Choices?
There are seven paintings to choose from.
That means that we count down from the seven just as before. Only we’re limited by the three spaces. That gives us:
7 x 6 x 5
Does Order Matter?
Last but not least…
Remember, we’re looking for the different orders (or different ways) that the paintings can be organized. So… er… yeah–order matters. That means we just keep it the way it is, because the original Factorial calculation tells us that order matters.
That gives us:
7 x 6 x 5 = 210
If it helps, that means the order Painting A, Painting B, Painting C would be considered different from the order Painting B, Painting A, Painting C and so forth even though we have selected the same paintings.
All that said, if we wanted to find the situation where order doesn’t matter, this would mean that Painting A, Painting B, Painting C would be considered the same as Painting B, Painting A, Painting C and so forth.
This is getting slightly ahead, so let’s move on and catch up with ourselves. Next topic: Combinations.
Chapter 3: Combinations on the GMAT
Combinations are simply Permutations where the order is randomized. In other words, the order doesn’t matter.
Three paintings, from a group of seven total paintings, are chosen to be hung on a wall. How many groups of three paintings can be chosen?
Think about it this way: we have calculated everything thus far assuming that the order does matter. In fact, simply by virtue of the fact that we are using Factorials to build these circumstances we must build everything using situations where order matters.
A Combination is, essentially (and precisely), a Permutation where we have randomized the orders of the chosen objects. You would be correct and deserve a cookie if you noticed that we are actually doing extra work to make the situation simpler. If that seems a bit stupid, well, it is. So much of math could be more efficient, eh?
Let’s work this out with the above situation.
The Combination would be asking how many groups of three paintings could be selected (rather than how many ways three paintings could be selected). That means that if we pick Painting A, Painting B, and Painting C, that would be the same group as Painting B, Painting A, Painting C and so forth.
In other words:
abc = acb = bac = bca = cab =cba
= 3! possibilities
That means six different situations are counted as the same, so we really just need to count one for every six. We get the value six by taking the factorial of the size of the group, in this case 3!.
That is, the Combination in our sample case becomes this:
Note that the 6 and the 3! Cancel each other easily because 3! = 6.
Really, as we go on from here, knowing that 3! = 6 and 4! = 24 and 5! = 120 will save you a lot of grief; it would be useful to commit these to memory.
IMPORTANT NOTE: the situation where you need to specifically know the order will be quite rare. You can think about it as situations where you “name the spaces,” e.g., Person A as President and Person B as Secretary would be different from Person A as Secretary and Person B as President.
If you really need to take a guess–and after all, if you ask yourself the Core Questions, the distinction between Permutation and Combination will be clear–then the vast majority of the time you’ll be asked to find the number of different possibilities that constitute a group. That is, a Combination.
In short, if you must, punt on the side of Combinations.
GMAT Combinations: How to Use Shorthand
Because it’s a pain to write all the math all the time when one is planning in GMAT Combinatorics questions, the “shorthand versions” or Permutation and Combination are often used.
For our case, the shorthand Permutation would be this:
7P3 = 7*6*5
The shorthand Combination would be this:
In each case, note that the number of Choices is on the left and the number of Spaces is on the right. In the case of the Combination, the number of Spaces is the same as the Factorial on the bottom.
In other words, we can say that a Combination is simply a Permutation divided by the Factorial of the group size. That means we could even choose to write the shorthand this way:
Note: different shorthand systems write the Permutation and Combination in different ways. You’ll often see them written like this, for example:
Ultimately, you won’t see any of this shorthand on the GMAT–remember, you’re just using it to plan those complicated-ass problems–so you could really pick any shorthand system and it wouldn’t make any difference. I prefer to write in the nPr and nCr form, but you can do whatever the hell you like.
To recap, let’s look at another brief example, proposed in a slightly different way.
Chapter 4: GMAT Permutations and GMAT Combinations Recap
Let’s consider a case where we have six different-colored marbles: red, blue, green, yellow, purple, and chartreuse.
If we were to put four of these marbles into a box, we could select four and lay them directly into the box.
How Many Spaces?
Four fit in the box: _ _ _ _
How Many Choices?
We begin with six: 6 x 5 x 4 x 3
Now, we still don’t have enough information to know whether we’re looking at a Permutation or a Combination.
We have put the marbles in a specific order. That is, we’d treat Red, Blue, Green, and Yellow as a different order from Blue, Red, Green, Yellow, and so forth. There will be 4! = 24 ways that this set of four colors could be distributed.
The Permutation counts all of these possibilities as different.
Think about it this way: we have, of course, put the marbles into the box in a particular order. At this point, they’re just sitting there in that same order. The “different orders” refers to the number of different ways that we could have placed them.
Here is our Permutation:
6 x 5 x 4 x 3 = 360
That’s where dividing by the factorial of the group size comes in. That means that we only care about the different groups of 4. We therefore have a factor of 4! = 24 fewer possibilities than we calculated with the Permutation.
In other words, we divide by 4! because we have four spaces. You can think about this as picking the box up and shaking it. Unless you’re very, very lucky, I think we can assume that this action will put the four marbles in a different order. In other words, it randomizes the order.
To yield the Combination, then, we see:
(6*5*4*3)/4! = (6*5*4*3)/(4*3*2*1) =5*3 =15
If you’re still skeptical, you can double-check by taking the Permutation and dividing by 4! = 24.
So it does work after all!
Learn more about GMAT Combinatorics problems in the Ultimate Guide to GMAT Combinatorics here!