Math Things to Memorize: the Combinations Formula

For some questions on the GMAT, you will need to know the Combinations formula.

Combinations formula = n! / k! (n-k)!

n = the bigger number (what we’re choosing from) 
k = the smaller number (how many we’re choosing)

Check out this question from Math Revolution:

How many committees can be formed comprising 2 male members selected from 4 men, 3 female members selected from 5 women, and 3 junior members selected from 6 juniors?

A. 900
B. 1200
C. 1500
D. 1800
E. 2400

We could almost rephrase this as three different questions:

How many ways to choose 2 from 4?
How many ways to choose 3 from 5?
How many ways to choose 3 from 6?

Let’s start with the first question: how many ways to choose 2 from 4? 

4! / 2! (4-2)!
4! / 2!2!
(4 x 3 x 2 x 1) / (2 x 1)(2 x 1)
24/4 = 6

But if we counted this and said the four males were ABCD, we could just list AB, AC, AD, BC, BD, and CD, and you can see it also equals 6.

Anyway, finding the other two:

How many ways to choose 3 from 5?

5! / 3! (5-3)!
5! / 3! 2!
We can cancel out 3! from both numerator and denominator.
5 x 4 / 2
20/2 = 10

How many ways to choose 3 from 6?

6! / 3! (6-3)!
6! / 3! 3!
We can cancel out one 3! from both numerator and denominator.
6 x 5 x 4 / 3 x 2
120 / 6 = 20

Back to our original questions:

How many ways to choose 2 from 4? 6
How many ways to choose 3 from 5? 10
How many ways to choose 3 from 6? 20

Now, multiply all those numbers together! :)

6 x 10 x 20 = 1200

The correct answer is (B).

Backsolving Problem Solving “Work” Questions

One way to do these type of question is to Backsolve, or essentially “try out” the answer choices. Let’s look at a problem:

Working together, each at his or her own constant rate, Jeff and Ashley painted their apartment in 6 hours. Working at his constant rate, Jeff could have painted the whole apartment in 10 hours. How many hours would it have taken Ashley, working at her constant rate, to paint the apartment?

A. 4
B. 12
C. 15
D. 16
E. 20

Let’s say it takes Ashley 15 hours (choosing answer choice (C)). Ashley would have a 1/15 rate and Jeff’s rate is 1/10 each hour. Working together they would do 1/15 + 1/10 each hour, or 2/30 + 3/30 = 5/30 = 1/6 in one hour. Then, yes, it does make sense that together the would do the job in 6 hours.

We got lucky here that (C) ended up being the correct answer, but if our answer had not matched the given information, then we probably could have discerned the correct answer based on how “too big” or “too small” we were.

While not required, sometimes we forget how useful leveraging the answer choices can be! :)