Question

of 19 possible books, you plan to take 5 with you on vacation. how many different collections of 5 books can you take?
you can take \square different collections of 5 books on vacation with you.

Explanation:

Step1: Identify the combination formula

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to be chosen. Here, \(n = 19\) and \(r=5\).

Step2: Calculate factorial values

First, calculate \(n!=19! = 19\times18\times17\times16\times15\times14\times\cdots\times1\), \(r!=5! = 5\times4\times3\times2\times1=120\), and \((n - r)!=(19 - 5)!=14! = 14\times13\times\cdots\times1\). Then \(C(19,5)=\frac{19!}{5!(19 - 5)!}=\frac{19\times18\times17\times16\times15\times14!}{5!\times14!}\).

Step3: Simplify the expression

Cancel out the \(14!\) terms. We get \(C(19,5)=\frac{19\times18\times17\times16\times15}{5\times4\times3\times2\times1}\).

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Answer:

\(11628\)