In how many ways can we distribute 5 distinct balls, B1,B2,...,B5 in 5…
2004
In how many ways can we distribute 5 distinct balls, B1,B2,...,B5 in 5 distinct cells, C1,C2,...,C5 such that Ball B,is not in cell C,Vi=1,2,...,5 and each cell contains exactly one ball?
- A.
44
- B.
96
- C.
120
- D.
3125
Attempted by 5 students.
Show answer & explanation
Correct answer: A
Answer: 44
Explanation using inclusion–exclusion (derangement formula):
Total permutations: 5! = 120.
Number of permutations with no ball in its original cell (derangements) is given by
!5 = 5! × (1 - 1/1! + 1/2! - 1/3! + 1/4! - 1/5!)
Compute the alternating sum: 1 - 1 + 1/2 - 1/6 + 1/24 - 1/120 = 0.366666... , so !5 = 120 × 0.366666... = 44.
Therefore there are 44 valid distributions.
Alternative quick check using recurrence:
Use !n = (n-1)(!(n-1) + !(n-2)) with !1 = 0 and !2 = 1.
Compute: !3 = 2, !4 = 9, !5 = (5-1)(!4 + !3) = 4 × (9 + 2) = 44.