Question

Twenty people check their hats at a theater. In how many ways can their hats be returned so that

(a) no one receives his or her own hat?

(b) at least one person receives his or her own hat?

(c) exactly one person receives his or her own hat?

Answer 1

GIVEN THAT:

Twenty people check their hats at a theater. In how many ways can their hats be returned so that

(a) no one receives his or her own hat?

(b) at least one person receives his or her own hat?

(c) exactly one person receives his or her own hat?

1. (a) The number of people is 20, thus the number of hats is 20. Number of ways that no one receives his or her own hat is given by the derangement of 20 hats that is.

2. We know thatwhere n is any natural number, is given by

   

3. Therefore we have,

   

4. Hence number of ways that no one receives his or her own hat is

   .

5. (b) Number of all permutation of 20 hats is and since the number of ways that no one receives his or her own hat is. Therefore the number of ways that at least one person receives his or her own hat is given by

   .

6. (c) Since there are 20 choices for the person who receives his or her own hat. For each choice, the 19 other people can get their hats in derangement of 19 hats that is.Hence the number of ways that exactly one person receives his or her own hat is given by

   .