Question

If $n$ objects are distributed at random among $n$ persons, the probability that at least one of them will not get anything is

• A. $1–\frac{(n-1)!}{n^{n-1}}$
• B. $\frac{(n-1)!}{n^n}$
• C. $1–\frac{(n-1)!}{n^n}$
• D. None of these

Answer 1

The correct option is C

$1–\frac{(n-1)!}{n^n}$

Explanation for correct option :

Probability:

We know that Probability of an event $=\frac{\mathrm{number}\mathrm{of}\mathrm{favorable}\mathrm{outcomes}}{\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$

The first object can be distributed to any of the $n$ persons.

So the total number of ways of distributing the n objects randomly among $n$ persons is $n^n$

There are${}^n{P}_n=n!$ ways in which each person gets exactly one object.

Therefore probability that each person will gets exactly one object $=\frac{n!}{n^n}$

$=\frac{(n-1)!}{n^{n-1}}$

Hence the probability that at least one person does not get any object =$1–\frac{(n-1)!}{n^{n-1}}$

Hence option C is the answer.