How many distinct permutations of the string "NADAMADRID" have the word DAM appearing in them? How many distinct permutations of the string "NADAMADRID"  have the word DAM appearing in them?
Normally, under the Mississippi Rule, you would take the total number of characters factorial, then divide by the product of all the characters that repeat factorial. In this case however, they ask how many times a certain word will appear in the permuations of a bigger string. I was confused on how to do this problem, and how i would count these possibilites.

If we discard the word $DAM$ temporarily, we have $7$ letters to arrange, out of which two letters of two types are similar. The arrangement of these can be done in $\frac{7!}{(2!)^2}$ ways. Once this is done, we have $8$ spaces for the word $DAM$ to be inserted into , so in total you get $$\frac{7!}{(2!)^2} \times 8 = 10080$$ ways.
