To determine how many different 5-book selections are possible from a list of 15 titles, we use the concept of combinations. Combinations are used when the order of selection does not matter.
The formula for combinations can be written as:
[tex]\[ C(n, k) = \frac{n!}{k!(n - k)!} \][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of items to choose from,
- [tex]\( k \)[/tex] is the number of items to choose,
- [tex]\( n! \)[/tex] represents the factorial of [tex]\( n \)[/tex], and
- [tex]\( k! \)[/tex] represents the factorial of [tex]\( k \)[/tex].
Given the problem:
- [tex]\( n = 15 \)[/tex] (total number of titles),
- [tex]\( k = 5 \)[/tex] (number of titles to choose).
Substituting these values into the formula, we get:
[tex]\[ C(15, 5) = \frac{15!}{5!(15 - 5)!} = \frac{15!}{5! \cdot 10!} \][/tex]
After performing the calculations (which we can infer generates factorial values and divisions), we find that the number of ways to choose 5 books from 15 titles is:
[tex]\[ C(15, 5) = 3,003 \][/tex]
Therefore, there are 3,003 different possible selections of 5 books from the list of 15 titles.
