Determine the number of strings of length five consisting of five distinct capital letters (A-Z)

Question

A- that do not contain an A.

B- that contain an A.

C- that contain an A, a B, a C as their first three symbols, in that order.

D- that contain an A, a B, and a C as their first three symbols, in any order.

E- that contain an A, a B, and a C somewhere, in that order.

F- that contains an A, a B, a C somewhere, in any order.

How I thought of this questions is, there are 26 letters (A-Z), so what makes it not contain an A is that we don't know which capital letters they're talking about, so any capital letters, by repetition we have 52 letters containing capital and non capital letters. Without (A). 51C5= 2349060. Contains an (A)= 2598960 I don't know if I'm doing this right, and not really where to start for the rest.

Any help will be appreciated

Answer 1

Without loss of generality you assume only for capital letters.

For part C you have $21*20$ choices.

For part D you have $21*20 *(3!)$ choices.

For part E you have $21*20 *(6)$ choices.

For part F you have $21*20 *(5*4*3)$ choices.