Question

How many ways are there to place 5 flowers into 3 identical bins (such that no bin remains empty)?

• A. 27
• B. 33
• C. 36
• D. 25

Answer 1

The correct option is D 25

A Stirling number of the second kind, denoted as S(n, r), is the number of ways a set of n elements can be partitioned into r non-empty sets.
The goal is to find S(5, 3). From the recurrence relation,
S(5,3) = 3S(4,3) + S(4,2)
The values of S(4,3), S(4,2) can be drawn from the basic cases. $S(4,3){=}^4{C}_2=6$ and $S(4,2)=2^{4-1}-1=7$.
Thus, S(5,3) = 3 $\ast$ 6 + 7 = 25.