1. ## counting question

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?

2. I know this is a derangement question,
so would this be right for part a ?

d20 * d20 ??

3. Originally Posted by sbankica
(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?
The part (a) simple. If D(20) is the number of derangements of twenty items then answer to part (a) is $D(20)^2$.

Part (b) is another matter.
The number of ways that at least one person gets both his items returned is:
$\sum\limits_{k = 1}^{20} {\left( { - 1} \right)^{k+1} \binom{ 20}{k} \left[ {\left( {20 - k} \right)!} \right]^2 }$

Can you finish?