How many 3-letter words can we make from the letters A, B, C, and D, if we are allowed to repeat letters, and we must use the letter A at least once? (Here, a word is an arbitrary sequence of letters.)

In this case, it might be easier to count the number of total words and subtract the number of words that do not contain A.

There are 4*4*4=64 total words

There are 3 letters excluding A, so 3*3*3=27 words w/o A

64-27=37 possible words that contain A.