How many derangements are there of the word MATHEMATICAL?

Asked Mar 20 '17 at 14:28

Active Mar 20 '17 at 17:36

Viewed 217 times

I know how to get the number of derangements for a word that has no repeating letters for example "guitar":

$$6!\left(\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}\right)$$

I also know how to get the number of derangements of a word with one set of repeating letters, such as "bottle", as this has been answered in another post.

However, when there are MULTIPLE sets of repeating letters, as in "MATHEMATICAL" (2 M's, 2 T's, and 3 A's), it becomes a bit more complicated considering if the first M is in the second M's place, according to the formula it's a derangement, but it actually is not. Could anyone help me out with this? Thanks!

edited Mar 20 '17 at 17:36

asked Mar 20 '17 at 14:28

Alex Laferriere

You know the formula. Can you derive the formula? – Antoine Mar 20 '17 at 14:32
$M^2, A^3, T^2, H^1, E^1, I^1, C^1, L^1$ The answer is: $65,488,752$ by the method of rook polynomials outlined in my answer to the BOTTLE question. To do this I used this tool which I also mention in my answer. – N. Shales Mar 20 '17 at 18:45
It's a duplicate just because it's is WORD. – GNUSupporter 8964民主女神地下教會 Mar 20 '17 at 22:20

How many derangements are there of the word MATHEMATICAL?

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