How many strings of length 5 of the letters A, B, C, D, and E contain exactly one A and exactly one D if repetition of letters is allowed?

We must have exactly 1 A and exactly 1 B  the other 3 letters can be C,D,E repeats allowed, so that is 3^3=27 ways with order counting.

 

The other A and B must be slotted into the string.        5C2=10 places but either the A or B can go first so that is 20 places.

 

So I get 27*20 = 540 ways.

AD, 1 of each letter, 5! ways to arrange: 120

AD, 2 of one letter, 1 of another, 6 possible choosing of letters, 60 ways to arrange: 360

AD, 3 of one letter, 3 possible ways to choose, 20 ways to arrange: 60

60+360+120

Total 540 ways.  :))

 

=^._.^=