Five distinct cars are to be placed in three identical showrooms. Each…
2024
Five distinct cars are to be placed in three identical showrooms. Each showroom can hold up to all five cars. In how many ways can the cars be placed so that no showroom remains empty?
- A.
20
- B.
24
- C.
25
- D.
10
Attempted by 1 students.
Show answer & explanation
Correct answer: C
Concept — Placing distinct objects into identical (unlabelled) groups so that no group is left empty is counted by the Stirling number of the second kind S(n, k): the number of ways to split a set of n distinct elements into k non-empty, unordered subsets.
Application — Here the n = 5 cars are distinct and the k = 3 showrooms are identical and must all be used, so the count is S(5, 3). Break it up by the group-size patterns that sum to 5 using exactly three non-empty parts:
Pattern 3-1-1: choose the 3 cars that share a showroom in C(5, 3) = 10 ways; the remaining two cars each sit alone, so this pattern gives 10 groupings.
Pattern 2-2-1: pick the single car in C(5, 1) = 5 ways, then split the remaining 4 cars into two unordered pairs in 3 ways, giving 5 x 3 = 15 groupings.
Total — S(5, 3) = 10 + 15 = 25.
Cross-check — Using the recurrence S(n, k) = k.S(n-1, k) + S(n-1, k-1): S(5, 3) = 3.S(4, 3) + S(4, 2) = 3.6 + 7 = 25, which matches.