If n is the number of ways in which 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 [2024]

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Solution

Q.
If $n$ is the number of ways in which 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 [2024]

1 43

2 47

3 53

4 51

Ans.
(4)

Number of ways in which five different employees can sit are

$I$    $I I$    $I I I$ $I V$

$□$ $□$ $□$ $□$

5 0 0 0    $\to$ 5!/5! = 1

4 1 0 0 $\to$ 5!/4! = 5

3 2 0 0 $\to$ 5!/3!2! = 10

3 1 1 0    $\to$ 5!/3! 1! 1! 2! = 10

2 2 1 0    $\to$ $\frac{5!}{2! 2! 1! 2!} = 15$

2 1 1 1 $\to$ $\frac{5!}{2! 1! 1! 1! 3!} = 10$

$∴$   Total number of ways =1 + 5 + 10 + 10 + 15 + 10 = 51

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