If a club has 12 members, what is the ratio GMAT

Hey folks, here’s the actual question text:

If a club has 12 members, what is the ratio of the number of different possible 6-member committees that can be formed to the number of different possible 5-member committees that can be formed?

a) 7 to 1

b) 6 to 1

c) 7 to 5

d) 6 to 5

e) 7 to 6

Congratulations! If you have memorized the Combination formula and desperately want to use it but are beginning to realize that (naturally) the GMAT actually expects you to think rather than plug and chug, it’s your lucky day.

Now of course this is GMAT and not some shitty weekly quiz that your tenth grade teacher wrote while drunk, so it obviously won’t be quite as simple as plug and chug.

Let’s dive in:

First, you’ll want to figure out what it’s actually asking. When you get a ratio question, you need to figure out what exactly the ratio is between.

In other words, what are the top and the bottom of the ratio in question?

Top:

Here, we want the number of possible 6-person committees that can be chosen from 12. That is, of course, the old “12 choose 6”:

12C6

Now I’ll suggest we start by doing this the actual way using slot method rather than doing it the brain-dead way with that execrable n!/(r!(n-r)!) Excel macro garbage that people have such a hard-on to apply. Are you taking GRE or GMAT?

That is,

Now just hold your horses and leave that until we figure out what the bottom is. DO NOT START CANCELING YET. It won’t be as much fun until we get the whole thing stacked up!

Bottom:

Here, we want the number of 5-person committees that can be selected from out of 12 people. The old “12 choose 5”:

12C5

Combining the two:

That leaves us with the following–you’ll notice that you can cancel the two denominators with each other and the two numerators with each other, like this:

(Otherwise you can simply flip and multiply as with a normal fraction, but where’s the fun in that?)

Then of course you’ll notice that everything now cancels down. We see that the only thing left is the 7/6 term.

The useless, execrable garbage method:

I’m only including this shit because it’ll help SEO, not to be nice. So don’t believe I’m doing it for your benefit. However maybe, in this one case ever that I’ve ever seen ever I can see a reason that one wouldn’t be totally fucking wasting time using these idiotic equations.

Let’s line up the equations, assuming…

Now on the surface it might seem that what we’re after here is opening up all of the factorials, which would be a ridiculous and petty waste of time.

Rather, kill the 12!, then flip:

Then of course you’ll notice that this becomes…

…once you realize that the only term on top that isn’t canceled by the bottom is 7 and that the only term on bottom that isn’t canceled by the term on top is 6.

If you want to be able to tackle Permutation and Combination questions confidently (or at least without soiling yourself), check out these resources:

–The Ultimate Guide to Permutations and Combinations

–The GMAT Permutations and Combinations Guide

If a club has 12 members , what is the ratio GMAT is the thing that I’m supposed to type over and over so you can ignore the part from here because it’s just to appease the robot overlords.

Rowan