How many distinct three-letter sequences with at least one $T$ can be formed by using three of the six letters of $TARGET$? One such sequence is $T-R-T$. [MathCounts 2005 National Countdown]
The problem given is easy, because the word given is small enough for casework. (One $T$ implies there are $3*4*3=36$ words. Two implies there are $3*4=12$).
However, how does this generalize to $n$ given characters and $m<n$ letter long distinct words? If $n=m$, there is the standard technique of dividing by the number of repeated letters factorial, but for a long word such as $AAABCCDEEEFFG$, I don't know how to effectively compute all possible $7$ letters words.
(All I need is this, as generalizing the "at least one $T$'s" part is just subtracting the words that can be made with zero $T$'s)