Question

of 12 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, \(n = 12\) and \(r=5\).

Step2: Calculate factorial values

First, calculate \(n!=12! = 12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1\), \(r!=5! = 5\times4\times3\times2\times1\), and \((n - r)!=(12 - 5)!=7! = 7\times6\times5\times4\times3\times2\times1\).
Then, \(C(12,5)=\frac{12!}{5!(12 - 5)!}=\frac{12\times11\times10\times9\times8\times7!}{5\times4\times3\times2\times1\times7!}\).

Step3: Simplify the expression

Cancel out the \(7!\) terms. Then, \(\frac{12\times11\times10\times9\times8}{5\times4\times3\times2\times1}\).
\(12\div4 = 3\), \(10\div5=2\), \(9\div3 = 3\).
The expression becomes \(3\times11\times2\times3\times8\).
\(3\times11 = 33\), \(33\times2=66\), \(66\times3 = 198\), \(198\times8=792\).

Answer:

\(792\)