Combinations
Learning Objectives
- Find the number of combinations of n distinct choices
- Second
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 ways to order 3 paintings. There are , 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 distinct objects, the number of ways to select objects from the set is
How To: Given a number of options, determine the possible number of combinations.
- Identify from the given information.
- Identify from the given information.
- Replace and in the formula with the given values.
- Evaluate.
Example: 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?
Answer:
- We want to choose 2 side dishes from 5 options.
- We want to choose 3 side dishes from 5 options.
Analysis of the Solution
We can also use a graphing calculator to find combinations. Enter 5, then press , enter 3, and then press the equal sign. The , 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 objects. Therefore, .Try It
An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?Answer: