Question

a librarian chooses seven holiday books from a selection of ten to be displayed in the window of the library. in how many different ways can she choose the group of seven books?

Explanation:

Step1: Identify the problem type

This is a combination problem since the order of choosing the books does not matter. The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 10 \) (total number of books) and \( k=7 \) (number of books to choose).

Step2: Substitute values into the formula

We have \( n = 10 \) and \( k = 7 \). First, note that \( C(n,k)=C(n,n - k) \), so \( C(10,7)=C(10,3) \). Calculating \( C(10,3)=\frac{10!}{3!(10 - 3)!}=\frac{10!}{3!7!} \).

Step3: Simplify the factorials

\( 10! = 10\times9\times8\times7! \), so substituting back we get \( \frac{10\times9\times8\times7!}{3!\times7!} \). The \( 7! \) terms cancel out. Then \( 3! = 3\times2\times1=6 \). So we have \( \frac{10\times9\times8}{6} \).

Step4: Calculate the result

\( 10\times9\times8 = 720 \), and \( 720\div6 = 120 \).

Answer:

120