Question

If the number of ways in which $n$ different things can be distributed among $n$ persons so that at least one person does not get any thing is $232$. Then $n$ is equal to

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (B)

$4$

The number of ways in which $n$ things are distributed among $n$ persons such that every person gets one thing is $n!$

The total number of ways of distributing $n$ things among $n$ people where each person can get any number of things is $n^n$

Therefore, the total number of ways in which $n$ things are distributed among $n$ persons such that at least one person gets nothing is $n^n-n!$

$\therefore n^n-n!=232$

There is no general method of solving this equation. By trial and error

$2^2-2!=2$

$3^3-3!=21$

$4^4-4!=232$

$\therefore n=4$

Option B is correct