Find the number of five-letter words from the three letter set A,B,C in which each letter occurs at least once

If each letter has to occur at least once , the possible combinations are :





1. Where any one letter occurs thrice , and the other two letters occur once each





The combinations will be A thrice , B and C once each , A and B once each , C thrice , and A and C once each , B thrice.





The number of such combinations is :





How can we select 3 positions for the letter which is repeated thrice , out of the available 5 slots ? This is 5!/(3! * 2!) = 10 ways. The remaining two slots can be taken by each of the other two letters in two different ways , thus giving a total of 20 different words in each case , for a total of 60 different words.








2. Where any two letters occur twice each , and the third letter occurs once.





The combinations will be A and B twice each , C once , A and C twice each , B once , and B and C twice each , A once.





The number of such combinations is :





How do we select 2 slots out of 5 for the first letter which is repeated twice ? This is 5!/(2! * 3!) = 10. Having selected these two slots , we are left with 3 slots , from which we can again select 2 slots for the second letter which is repeated twice. The number of ways this can be done is 3!/2! = 3. Thus the total number of ways = 10 * 3 = 30.





Since there are 3 different combinations in this category , we end up with 90 different words.





The total = 60 + 90 = 150.

Find the number of five-letter words from the three letter set A,B,C in which each letter occurs at least once
3! * 5C3 * 3 * 3 = 540
Reply:There are 3^5=243 possbile words with 3 letters


simply list them out and strike out the ones that do not contain each letter at least once!