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

The Correct Option is C

The number of distinct arrangements for partitioning the integer 5 into 4 components is enumerated as follows:

• Partition: 5, 0, 0, 0. Count: 1 arrangement.
• Partition: 4, 1, 0, 0. Count: \(\frac{5!}{4!} = 5 \text{ ways}\).
• Partition: 3, 2, 0, 0. Count: \(\frac{5!}{3!2!} = 10 \text{ ways}\).
• Partition: 2, 2, 1, 0. Count: \(\frac{5!}{2!2!1!} = 15 \text{ ways}\).
• Partition: 2, 1, 1, 1. Count: \(\frac{5!}{2!1!1!1!} = 10 \text{ ways}\).
• Partition: 3, 1, 1, 0. Count: \(\frac{5!}{3!1!1!} = 10 \text{ ways}\).

Summation:

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