Question

There are two copies of the letter A and an unlimited supply of the letters B, C, and D. How many words of length 10 can you make from these, such that: both A's are used and the 3rd letter is an A and there is no B appearing between the A's ? Hint: Consider the two cases: (i) the A in the third position is the first A in the word, and (ii) it is the second A.

Answer 1

(i) The A in the third position is the first A in the word:

Case 1:- If Second A position on 4th place:

No. of ways 3*3*1*1*3*3*3*3*3*3 =6561

Case 2:- If Second A position on 5th place:

No. of ways 3*3*1*2*1*3*3*3*3*3 =4374

Case 3:- If Second A position on 6th place:

No. of ways 3*3*1*2*2*1*3*3*3*3 =2916

Case 4:- If Second A position on 7th place:

No. of ways 3*3*1*2*2*2*1*3*3*3 =1944

Case 5:- If Second A position on 8th place:

No. of ways 3*3*1*2*2*2*2*1*3*3 =1296

Case 6:- If Second A position on 9th place:

No. of ways 3*3*1*2*2*2*2*2*1*3 =864

Case 7:- If Second A position on 10th place:

No. of ways 3*3*1*2*2*2*2*2*2*1 =576

(ii) The A in the third position is the second A in the word:

Case 8:- If first A position on 1st place:

No. of ways 1*2*1*3*3*3*3*3*3*3 =4374

Case 9:- If first A position on 2nd place:

No. of ways 3*1*1*3*3*3*3*3*3*3 =6561

TOTAL WAYS = Case 1+2+3+4+5+6+7+8+9 =29466 ways