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Math Help - Combinatorics question involving summing digits of numbers

  1. #1
    Junior Member CuriosityCabinet's Avatar
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    Combinatorics question involving summing digits of numbers

    How many five-digit numbers are there who digits sum to 39?

    Can someone give me their answer so I can check if mine is right? Thank you!
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    Quote Originally Posted by CuriosityCabinet View Post
    Can someone give me their answer so I can check if mine is right? Thank you!
    Why not post your answer as well as the method used to get it?
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    Junior Member CuriosityCabinet's Avatar
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    The different digit combinations possible are of the form:
    99993
    99984
    99975
    99885
    98886
    99966
    88887
    99876
    99777
    98877

    Then I worked out how many combinations of each digit there are and summed it, to get 210. But have a feeling that isn't quite right.
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    Quote Originally Posted by CuriosityCabinet View Post
    The different digit combinations possible are of the form:
    Then I worked out how many combinations of each digit there are and summed it, to get 210. But have a feeling that isn't quite right.
    I used a generating function and it gave the same answer.
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    Junior Member CuriosityCabinet's Avatar
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    Quote Originally Posted by Plato View Post
    I used a generating function and it gave the same answer.
    Can you explain to me your method?
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    Quote Originally Posted by CuriosityCabinet View Post
    Can you explain to me your method?
    One expands \left( {\sum\limits_{k = 1}^9 {x^k } } \right)\left( {\sum\limits_{k = 0}^9 {x^k } } \right)^4 .

    The coefficient of x^{39} is the answer.

    Notice the first sum starts at k=1 because the first digit in a five digit number cannot be zero.

    Here is an online tool you can use.
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