Question

How many different seven-letter sequences can be formed from the letters a, a, a, a, a, b, c?

Answer 1

If b is in the first position then c can be in any 1 of the remaining 6 positions.

If we start with ab then the letter c can be in any one of 5 positions and  if we have aab there are 4 possible positions for c and so on.

So the total number of possible sequences where b comes first = 6+5+4+3+2+1 =  21.

The same argument applies when c comes before b  so that gives us 21 ways also.

So the answer is 2 *21 = 42 different sequences.

A more direct way of doing  this  is to use factorials:-

answer = 7! / 5!     =  7 * 6 = 42.

 ( We divide by 5!   because of the 5 a's.)