To determine how many ways 7 books can be arranged on a shelf 5 at a time, we need to calculate the number of permutations of 7 books taken 5 at a time.
The formula for permutations is given by:
P(n, r) = n! / (n – r)!
Where:
- n = total number of items (in this case, 7 books)
- r = number of items to choose (in this case, 5 books)
Plugging in the values:
P(7, 5) = 7! / (7 – 5)! = 7! / 2!
This simplifies to:
P(7, 5) = (7 × 6 × 5 × 4 × 3 × 2 × 1) / (2 × 1) = 7 × 6 × 5 × 4 × 3
Now calculating that:
- 7 × 6 = 42
- 42 × 5 = 210
- 210 × 4 = 840
- 840 × 3 = 2520
Therefore, there are 2520 ways to arrange 7 books on a shelf 5 at a time.