Given the problem, we need to determine the number of ways to arrange 4 books out of 8 on a shelf. This is a permutation problem because the order in which the books are arranged matters.
The formula to calculate permutations is given by:
[tex]\[ P(n, r) = \frac{n!}{(n - r)!} \][/tex]
Where:
- \( n \) is the total number of items to choose from,
- \( r \) is the number of items to arrange.
In this problem:
- \( n = 8 \) (the total number of books),
- \( r = 4 \) (the number of books to be arranged).
Using the formula, we substitute the values:
[tex]\[ P(8, 4) = \frac{8!}{(8 - 4)!} \][/tex]
[tex]\[ P(8, 4) = \frac{8!}{4!} \][/tex]
Now, we need to calculate the factorials:
- \( 8! \) (8 factorial) is the product of all positive integers up to 8:
[tex]\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
- \( 4! \) (4 factorial) is the product of all positive integers up to 4:
[tex]\[ 4! = 4 \times 3 \times 2 \times 1 \][/tex]
Now, divide \( 8! \) by \( 4! \):
[tex]\[ P(8, 4) = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \][/tex]
The \( 4! \) in the denominator cancels out the \( 4! \) in the numerator, which simplifies to:
[tex]\[ P(8, 4) = 8 \times 7 \times 6 \times 5 \][/tex]
Therefore:
[tex]\[ P(8, 4) = 1680 \][/tex]
So, the number of ways to arrange 4 books out of 8 is:
[tex]\[ \boxed{1680} \][/tex]
