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Math Help - Discrete math problem with counting

  1. #1
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    Discrete math problem with counting

    Hi everyone, I have a problem whose solution I don't understand.

    Find the number of five-letter words that use letters from {A,B,C} in which each letters occurs at least once.
    Solution:
    3^5 -2^5 -2^5 -2^5 +1 +1 +1 = 150

    But, my solution is 3^5 -2^5 -2^5 -2^5 -1 -1 -1 = 144
    I don't know why I'm wrong. Please, help me to explain it. Thank you very much.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Discrete math problem with counting

    The +1's are to make up for the fact that the 2^5 terms double-count words AAAAA, BBBBB, and CCCCC. There are 2^5 ways to make up words using only A and B, including AAAAA and BBBBB. There are also 2^5 ways to make up words using only A and C, including AAAAA and CCCCC. And there are 2^5 ways to make up words using only B and C, including BBBBB and CCCCC. By subtracting 2^5 three times you have eliminated the possibility of AAAAA, BBBBB and CCCC twice each, so you need to add 3 to get the correct answer.
    Thanks from math88
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    Re: Discrete math problem with counting

    Quote Originally Posted by math88 View Post
    Hi everyone, I have a problem whose solution I don't understand.
    Find the number of five-letter words that use letters from {A,B,C} in which each letters occurs at least once.
    Solution:
    3^5 -2^5 -2^5 -2^5 +1 +1 +1 = 150
    That is the count or the number of surjections (onto functions) from a set of five to a set three.
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