How many anagrams of "DENOMBREMENT" exist?
The word "DENOMBREMENT" has $n=12$ letters, so one might initially think it has $n$! anagrams. However, among the $n$! ways to permute the letters, some are counted multiple times because certain letters appear more than once in the word. These letters are "E" which appears 3 times, "N" which appears 2 times, and "M" which also appears 2 times. Thus, in each permutation of the letters, the same anagram is obtained by swapping the "E"s, the "N"s, or the "M"s among themselves. Therefore, the same anagram corresponds to

$3!$ (permutations of the "E"s) $\cdot$ 2! (permutations of the "N"s) $\cdot$ 2! (permutations of the "M"s) $=24$ permutations. The number of anagrams of "DENOMBREMENT" is therefore

$$
\frac{12!}{24}=19958400
$$