Combinations on the ACT and GRE

Today I would like to continue discussing more advanced methods for solving counting problems.

Two days ago we discussed some more basic methods for solving counting problems on the ACT and GRE. These methods can be viewed by clicking the following link: Listing and the Counting Principle

And yesterday we talked about permutations. This article can be viewed here: Permutations 

Today I want to focus on another more advanced counting technique – Combinations.

Combinations

A combination is just a subset containing a specific number of the elements of a particular set. The number of combinations of n things taken r at a time is

For example, the number of combinations of {1, 2, 3} taken 2 at a time is

These combinations are 12, 13, and 23.

Note that 21 is the same combination as 12 (but 12 and 21 are different permutations). More on this below…

Just like the permutation formula, you do not need to know the combination formula for the ACT. You can do this computation very quickly on your TI-84 graphing calculator. To compute ₃C₂, type 3 into your calculator, then in the Math menu scroll over to Prb and select nCr (or press 3). Then type 2 and press Enter. You will get an answer of 3.

Difference Between Permutations And Combinations

Let me quickly tell you the difference between permutations and combinations: “Permutation” is just a fancy word for arrangement. Whenever you are counting the number of ways to arrange things you can use the permutation formula that I shared with you last week (or better yet the nPr feature in your calculator). So if you want to “order” things, “stack” things, “rearrange” things, choose specific tasks for people, etc, you would use permutations.

“Combinations” are used when you want to group things together without putting them in any specific order. This happens when you split a group of objects into two piles, or form a committee of people, for example.

So the numbers 12 and 21 are different permutations, but the same combination.

Here is a very basic combination problem:

Example 1

How many committees of 4 people can be formed from a group of 9?

The order in which we choose the 4 people does not matter. Therefore this is the combination ₉C₄= 126.

Here is our first standardized test math problem where combinations can be used:

Example 2

A chemist is testing 5 different liquids. For each test, the chemist chooses 3 of the liquids and mixes them together. What is the least number of tests that must be done so that every possible combination of liquids is tested?

First note that we can solve this problem by simply listing all of the possibilities. But be careful! This is just a bit tricky.

Solution by listing: In the following list a * means we are choosing that liquid, and an O means we are not:

***OO      *OO**
**O*O      O***O
**OO*      O**O*
*O**O      O*O**
*O*O*      OO***

We see that there are 10 combinations.

Alternatively, we can obtain the solution quicker by using what we have learned about combinations:

Solution using combinations:  We are counting the number of ways to choose 3 of the 5 liquids. This is ₅C₃ = 10.

And here is a more difficult standardized test math problem that can be solved quickly using combinations:

Example 3

Any 2 points determine a line. If there are 18 points in a plane, no 3 of which lie on the same line, how many lines are determined by pairs of these 18 points?

We need to count the number of ways to choose 2 points from 18. This is the combination ₁₈C₂ = 153.

Let me end this post by saying that it is unlikely that you will see a problem on a standardized test where you absolutely NEED to use permutations or combinations. You can almost always do counting problems either by forming a careful list, or by using the counting principle. But every now and then using permutations or combinations will get you the answer much more quickly.

Where To Find More Practice Problems

If you are preparing for the ACT, GRE or any other standardized test that has counting problems, you may want to take a look at the Get 800 collection of test prep books. Click on the picture below for more information.

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