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 :

Solution

Maths sir · 3 steps

Identical offices means only the groups matter, so list every way to split 5 people into at most 4 groups.

1. What to count. The offices look the same, so swapping the people of two offices gives the same arrangement. We only count how the 5 employees are split into groups, with at most 4 groups because empty offices are allowed.
2. Watch equal groups. When two groups have the same size, choosing them in either order gives the same split. So divide by 2! for each pair of equal-sized groups.

   Group sizes	Ways	Count
   5	everyone together	1
   4,1	C(5, 4)	5
   3,2	C(5, 3)	10
   3,1,1	C(5, 3), the two singles are fixed	10
   2,2,1	1/2!C(5, 2)C(3, 2)	15
   2,1,1,1	C(5, 2)	10

3. So. n = 1 + 5 + 10 + 10 + 15 + 10 = 51, which is option (c).

The idea to keep. For distinct objects into identical boxes, list the size patterns of the groups and count each one. Divide by k! whenever k groups have the same size.

Answer from the NTA final answer key. Solution by PrepWiser.