Question

Four men have $10$ hats. In how many ways can they wear them?

• A. $7!$
• B. $6!$
• C. $5400$
• D. $3040$

Answer 1

The correct option is A $7!$

It is given that four persons have $10$ hats to wear.

It can also be seen as there are $10$ different things out of which we need $4$ of them and do the arrangements of those things.

And, we know that number of ways of arranging $n$ different things taken $r$ at a time is nothing but ${}^n{P}_r$

So, according to question number of ways of wearing hats is ${}^{10}{P}_4$
$\Rightarrow {}^{10}{P}_4=\frac{10!}{(10-4)!}=\frac{10\times 9\times 8\times 7\times 6!}{6!}$
$\Rightarrow {}^{10}{P}_4=10\times 9\times 8\times 7=5040.$