We are given 5 distinct balls B_1, B_2, B_3, B_4, B_5 and 5 distinct cells C_1, C_2, C_3, C_4, C_5, and each cell contai...

Question

We are given 5 distinct balls B_1, B_2, B_3, B_4, B_5 and 5 distinct cells C_1, C_2, C_3, C_4, C_5, and each cell contains exactly one ball. In how many ways can the balls be placed in the cells such that B_i is not in cell C_i for 1 <= i <= 5?

A. 44
B. 96
C. 120
D. 3125

Accepted Answer

44 This is the classic derangement problem. We want permutations of 5 elements where no element appears in its original position. The number of derangements D(n) can be computed using the inclusion-exclusion formula. For n = 5, D(5) = 44.

We are given 5 distinct balls B_1, B_2, B_3, B_4, B_5 and 5 distinct cells C_1, C_2, C_3, C_4, C_5, and each cell contains exactly one ball. In how many ways can the balls be placed in the cells such that B_i is not in cell C_i for 1 <= i <= 5? A. 44 B. 96 C. 120 D. 3125