Question

A new employee checks the hats of 5 people at a restaurant, forgetting to put claim check numbers on the hats. When customers return for their hats, the checker gives them back hats chosen at random from the remaining hats. a) How many derangements of the hats are there

Answer 1

Answer:

There are 44 derangements of the hats.

Step-by-step explanation:

Solution:-

- A derangement is a permutation of objects that leaves no object in its original position.

- The permutation 21453 is a derangement of 12345 because no number is left in its original  position. However, 21543 is not a derangement of 12345, because this permutation leaves 4 fixed.

- Let Dn denote the number of derangements of n objects:

The number of derangements of a set with n elements is:

            $D_n = n!* [ 1 - 1/1! + 1 / 2! - 1 /3! + ...+ (-1)^n 1/n! ]$

- The answer is the number of ways the hats can be arranged so that there is no hat in its original  position divided by n!, the number of permutations of n hats:

            $D_5 = 5!* [ 1 - 1/1! + 1 / 2! - 1 /3! + 1 /4! - 1/5!]\\\\D_5 = 44 ways$

- There are 44 derangements of the hats.