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.

Just treat $DAM$ like another unique letter.  Then you have 8 letters total, with two $A$'s and two $D$'s (after you take out those used in $DAM$).  So the total number of arrangements is $$\frac{8!}{2!2!}$$
Note this would get more complicated if there were more than one $M$, as then you'd have to account for instances with two $DAM$'s by inclusion-exclusion.
