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Chapter 8: Problem 78

What is a combination?

Short Answer

Expert verified
A combination is a method of selecting items from a larger set where the order does not matter. The number of combinations can be found using the formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).

Step by step solution

01

Understanding Combination

A combination is a way of selecting items from a larger set. What makes combinations unique is that the order of the items selected does not matter. For example, if you have a set of three items - A, B, and C. A combination of two items could be AB, AC, or BC. In combinations, AB and BA are considered the same because the order does not matter.
02

Combination Formula

The formula to calculate combinations is given by: \( C(n, r) = \frac{n!}{r!(n-r)!} \) \nWhere: \n \( C(n, r) \) is the number of combinations. \n \( n \) is the total number of items. \n \( r \) is the number of items to select. \n\( ! \) means factorial, which is the product of all positive integers up to that number. For example, \( 5! = 5 脳 4 脳 3 脳 2 脳 1 \)
03

Using the Combination Formula

To use the formula, you simply substitute your total number of items for \( n \) and the number of items to select for \( r \). For example, if you have a set of 5 items (n=5) and you want to find out how many ways you can select 3 items (r=3), you would calculate \( C(5, 3) = \frac{5!}{3!(5-3)!} \)

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