last one for today. Thank you so much.

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?

 

How many ways can the other 2 letters be chosen where order does not count?

There are 4 ways that the letters an be the same i.e.      AA,  BB,  CC,  DD

and there are 4C2=6 ways if the letters are not the same. Making a total of 10 ways.

So there are 10 combinations of letters, how many permutations of these are there

AAA   1 way

AAB, AAC, AAD      3 ways each     3*3=9  ways

ABB, ACC, ADD      3 ways each     3*3=9 ways

ABC, ABD, ACD       3*2=6 ways each     3*6=18ways

 

1+9+9+18 = 37 ways