How many of the words contain the text string "mat" somewhere in the word?

The condition should be understood as "at least one $mat$" – $mat$ only has to appear somewhere and there is no restriction on the count of this substring.

Your answer for this part is correct and may be justified by the inclusion/exclusion principle. $7!$ is the number of permutations of letters when one of the $mat$s is considered a single "letter". We double-count the strings with two $mat$s here, so we subtract $\frac{5!}2$, the number of permutations when both $mat$s are considered "letters".