4. Twenty people attend a celebration at a fancy hotel. On the way in, each

person leaves a coat and an umbrella with the coat-check person. Over

the course of the event, he also does some celebrating and, at the end of

the evening, returns the umbrellas and coats in random order (each person

does receive a coat and an umbrella though).

(a) In how many ways can the coats and umbrellas be returned so that

no person gets back either of their possessions?

(b) In how many ways can the coats and umbrellas be returned so that

no person gets back both of their possessions?