The empty shelf can be filled in 120 ways
The total number of books is
[tex]n = 10[/tex]
The number of books to move is:
[tex]r = 3[/tex]
To fill the empty shelf with three books, we make use of the following combination formula
[tex]^{n}C_r = \frac{n!}{(n - r)!r!}[/tex]
So, we have:
[tex]^{10}C_3 = \frac{10!}{(10 - 3)!3!}[/tex]
Evaluate the expression
[tex]^{10}C_3 = \frac{10!}{7!3!}[/tex]
Expand
[tex]^{10}C_3 = \frac{10 \times 9 \times 8 \times 7!}{7! \times 3\times 2 \times 1}[/tex]
Cancel out the common factors
[tex]^{10}C_3 = \frac{10 \times 9 \times 8}{3\times 2 \times 1}[/tex]
Evaluate the products
[tex]^{10}C_3 = \frac{720}{6}[/tex]
Divide
[tex]^{10}C_3 = 120[/tex]
Hence, the empty shelf can be filled in 120 ways
Read more about combinations at:
https://brainly.com/question/11732255
