Sender: Karl Freitag

Please help me with this word problem. It is from a critical thinking course I am taking in college. The class is PHL/251. I believe the answer is 46656, but not sure. Could you tell me if it is right and if not how you came up with the answer.

Question: An anthroplogist discovers an isolated tribe whose written alphabet contains only six letters (call the letters A,B,C,D,E, and F). The tribe has a taboo against using the same letter twice in the same word. It is never done. If each different sequence of letters constitutes a different word in the language, what is the maximum number of six-letter words that the language can employ?

Please help me!

First ask yourself, how many one letter words are there? The answer is clearly 6. Each of these you can change to a two letter word by adding one of the 5 remaining letters. Thus there are 6x5 = 30 possible two letter words.

Similarly each of these two letter words can be extended to a three letter word by adding one of the 4 remaining letters. Hence there are 6x5x4 = 120 possible three letter words. I think you can complete the problem from here.

If there were no taboo then there would still be 6 possible one letter words. Each you could extend to a two letter word by adding any one of the 6 letters. Thus there would be 6x6 = 36 possible two letter words. Similarly each of these two letter words can be extended to a three letter word by addind any of the six letters. Thus there are 6x6x6 = 216 possible three letter words. Continuing in this fashion, with no taboo there are 6x6x6x6x6x6 = 6⁶ = 46656 possible six letter words.