Question

of 13 possible books, you plan to take 7 with you on vacation. how many different collections of 7 books can you take?

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, $n = 13$ and $r=7$.

Step2: Calculate factorial values

$n!=13! = 13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $r!=7!=7\times6\times5\times4\times3\times2\times1$, and $(n - r)!=(13 - 7)!=6!=6\times5\times4\times3\times2\times1$.
$C(13,7)=\frac{13!}{7!(13 - 7)!}=\frac{13!}{7!6!}=\frac{13\times12\times11\times10\times9\times8\times7!}{7!\times6\times5\times4\times3\times2\times1}$.

Step3: Simplify the expression

The $7!$ in the numerator and denominator cancels out. Then $\frac{13\times12\times11\times10\times9\times8}{6\times5\times4\times3\times2\times1}=\frac{13\times12\times11\times10\times9\times8}{720}=1716$.

Answer:

1716