So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with **combinations**. A selection of [latex]r[/latex] objects from a set of [latex]n[/latex] objects where the order does not matter can be written as [latex]C\left(n,r\right)[/latex]. Just as with permutations, [latex]\text{C}\left(n,r\right)[/latex] can also be written as [latex]{}_{n}{C}_{r}[/latex]. In this case, the general formula is as follows.

An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are [latex]3!=3\cdot 2\cdot 1=6[/latex] ways to order 3 paintings. There are [latex]\frac{24}{6}[/latex], or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are *not* selecting 1 painting. There are 4 paintings we could choose *not* to select, so there are 4 ways to select 3 of the 4 paintings.

### A General Note: Formula for Combinations of *n* Distinct Objects

Given [latex]n[/latex] distinct objects, the number of ways to select [latex]r[/latex] objects from the set is

### How To: Given a number of options, determine the possible number of combinations.

- Identify [latex]n[/latex] from the given information.
- Identify [latex]r[/latex] from the given information.
- Replace [latex]n[/latex] and [latex]r[/latex] in the formula with the given values.
- Evaluate.

### Example 4: Finding the Number of Combinations Using the Formula

A fast food restaurant offers five side dish options. Your meal comes with two side dishes.

- How many ways can you select your side dishes?
- How many ways can you select 3 side dishes?

### Solution

- We want to choose 2 side dishes from 5 options.
[latex]\text{C}\left(5,2\right)=\frac{5!}{2!\left(5 - 2\right)!}=10[/latex]
- We want to choose 3 side dishes from 5 options.
[latex]\text{C}\left(5,3\right)=\frac{5!}{3!\left(5 - 3\right)!}=10[/latex]

### Analysis of the Solution

We can also use a graphing calculator to find combinations. Enter 5, then press [latex]{}_{n}{C}_{r}[/latex], enter 3, and then press the equal sign. The [latex]{}_{n}{C}_{r}[/latex], function may be located under the MATH menu with probability commands.

### Q & A

### Is it a coincidence that parts (a) and (b) in Example 4 have the same answers?

*No. When we choose r objects from n objects, we are not choosing [latex]\left(n-r\right)[/latex] objects. Therefore, [latex]C\left(n,r\right)=C\left(n,n-r\right)[/latex]. *

### Try It 8

An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?