In how many ways can 12 people be seated on a bench if only 5 spaces are available ?

The number of ways 12 people can be seated on a bench with only 5 spaces available is calculated using the permutation formula.

P(n, r) = n! / (n - r)!

Where:
n = total number of people (12)
r = number of available spaces (5)

P(12, 5) = 12! / (12 - 5)!
P(12, 5) = 12! / 7!
P(12, 5) = (12 x 11 x 10 x 9 x 8 x 7 x 6) / (7 x 6 x 5 x 4 x 3 x 2 x 1)
P(12, 5) = 95,040

Therefore, there are 95,040 ways in which 12 people can be seated on a bench with only 5 spaces available.

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