Principle of I/E exercise

Question

$10$ ladies drop off their red hats at the hat check of the museum. As they are leaving, the check hat attendant gave their hats back randomly. In how many ways exactly $6$ of the ladies receive their own hats (and the others not)?

I guess applying the binomial coeffcient of $(10,6)$ and this multiplying with derangement of $4$ I can get the answer. The result is $1890$.

I don't know if I'm wrong.

Answer 1

Yes in order for $6$ ladies to be selected, there are $\binom{10}{6}$ ways and for the other $4$ ladies to each not get their own hat, there are $!4$ ways. $$\binom{10}{6} \times!4=210\times 9 =1890$$