Problem 4.29 Suppose some friends go to a party. They each wear a coat. However, as they are leaving, they each randomly grab a coat. How many ways can the friends leave so that none of them have their own coat, if there are 5 friends?

The week 4 solution says 76 computed as following

5!− 5! / 2! + 5! / 3! − 5! / 4! + 5! / 5! = 76.

But correct answer seems to be 5! - 76 = 44 .

76 is the number of ways at least one friend grabbed his/her own coat and 44 is the number of ways no one left with their own coat.

It's easy to verify there are only 2 ways when there are 3 friends which is NOT

3! - 3!/2! + 3!/3! =  4.  Instead 6 - 4 = 2 is correct.

Can someone confirm ?

Thanks.