Question

In how many ways the word "BOTTLE" can be arranged?​

Answer 1

Answer:

There are ⌊6!e⌋=265 derangements of a 6-element set, but applying that formula to the letters of "BOTTLE" has two problems:

Some of these "derangements" move the T in the fourth position to the third position, or vice versa, or both, so once we take that into account, they're no longer actual derangements.

Of the valid derangements, each is counted twice: if you switch the two T's, that's a different permutation of a 6-element set, but shouldn't be a different derangement of the letters of "BOTTLE".

We begin by fixing the first problem. There are three cases:

The false derangement actually swaps the two T's. This case is in bijection (by swapping the two T's) with permutations that fix both T's and derange everything else, so there are ⌊4!e⌋=9 of these.

The false derangement moves the first T to the second T's place, and the second T to somewhere other than the first T's place. This case is in bijection (by swapping the two T's) with permutations that fix the second T and derange everything else, so there are ⌊5!e⌋=44 of these.

Same as the previous case, but with the second T going to the first T's place; also 44 of these.

This leaves us with 265−44−44−9=168 actual derangements.

The second problem is easy to fix; now we can divide by 2 and get 84 as our final answer.