Answer:
There are 44 derangements of the hats.
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! ]](https://img.qammunity.org/2021/formulas/mathematics/high-school/8j70yperl3rezvs7by636qeht4j8ns9004.png)
- 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](https://img.qammunity.org/2021/formulas/mathematics/high-school/luf1s9iuw9pdzw1lwdyuoxw6fsogqe0kqu.png)
- There are 44 derangements of the hats.