determining the number of possible groups of items\na librarian chooses seven holiday books from a selection…

Answer:

120

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, $n = 10$ and $r=7$.

Step2: Calculate factorial values

$n!=10! = 10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $r!=7!=7\times6\times5\times4\times3\times2\times1$, and $(n - r)!=(10 - 7)!=3!=3\times2\times1$.

Step3: Substitute into the formula

$C(10,7)=\frac{10!}{7!(10 - 7)!}=\frac{10!}{7!3!}=\frac{10\times9\times8\times7!}{7!\times3\times2\times1}$.

Step4: Simplify the expression

The $7!$ terms cancel out. We have $\frac{10\times9\times8}{3\times2\times1}=\frac{720}{6}=120$.