Question

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to:

• A. 47
• B. 53
• C. 51
• D. 43

Answer 1

Answer: (C)

51

Answer 2

Total ways to partition 5 into 4 parts are:

• 5, 0, 0, 0 → 1 way

• 4, 1, 0, 0 → $\frac{5!}{4!} = 5 \text{ ways}$

• 3, 2, 0, 0 →$\frac{5!}{3!2!} = 10 \text{ ways}$

• 2, 2, 1, 0 →$\frac{5!}{2!2!1!} = 15 \text{ ways}$

• 2, 1, 1, 1 →$\frac{5!}{2!1!1!1!} = 10 \text{ ways}$

• 3, 1, 1, 0 → $\frac{5!}{3!1!1!} = 10 \text{ ways}$

Total:

$1 + 5 + 10 + 15 + 10 + 10 = 51 \text{ ways}$