Combinations
Learning Objectives
- Find the number of combinations of n distinct choices
- Second
[latex]\text{C}\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}[/latex]
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[latex]\text{C}\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}[/latex]
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: 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. [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
An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split?Answer: [latex]C\left(10,3\right)=120[/latex]
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CC licensed content, Shared previously
- Ex: Determine the Number of Ways 6 Books can be Selected from 9 Books (Combination). Provided by: OpenStax Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
- Question ID 7175, 7156. Authored by: unknown, mb Lippman,David, mb Sousa,James. License: CC BY: Attribution.