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Math Help - a+b+c = 10

  1. #1
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    a+b+c = 10

    i have a question,not being able to think out a answer
    let
    a+b+c = 10
    find number of whole number solutions to the above equation

    i can find out the answer through balls n urns concept ie no of various solutions is equal to 12C2.

    but suppose i dont make any distinction between a,b,c
    and just wanna know the no of 3 numbers which add up to 10
    for example
    for the above equation 1+2+7 = 10 is a solution
    n so is 2+1+7 = 10
    but i dont wanna distinguish between the two cases n wanna treat them as 1 case only {1,2,7}
    how many more sets are there..

    plz help
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  2. #2
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    for the above equation 1+2+7 = 10 is a solution
    n so is 2+1+7 = 10
    treat them as 1 case only {1,2,7}
    But of course they are different solutions. If those were three children and we are giving out 10 cookies then it would makes a real difference.
    You are correct, there are {12 \choose 2} ways to place ten indistinguishable objects into three different cells.
    But in addition to the examples you gave that number also includes 0+0+10, 0+10+0, and 10+0+0.
    Do you want to include the possibility of an empty cell?
    It appears that the problem you really want to solve is to find the number of partitions if the integer 10 into three or less summands.
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  3. #3
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    Quote Originally Posted by Plato View Post
    But of course they are different solutions. If those were three children and we are giving out 10 cookies then it would makes a real difference.
    You are correct, there are {12 \choose 2} ways to place ten indistinguishable objects into three different cells.
    But in addition to the examples you gave that number also includes 0+0+10, 0+10+0, and 10+0+0.
    Do you want to include the possibility of an empty cell?
    It appears that the problem you really want to solve is to find the number of partitions if the integer 10 into three or less summands.
    yes i m including the possibilty of empty cells..
    actually i want the answer so that i can anlyse better in some questions where its not imp for us to find out possible arrangements n stuff..but to find other things like the max lcm or min lcm of the 3 numbers which add to 10..that ways
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  4. #4
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    Quote Originally Posted by ramanujam View Post
    yes i m including the possibilty of empty cells..
    actually i want the answer so that i can anlyse better in some questions where its not imp for us to find out possible arrangements n stuff..but to find other things like the max lcm or min lcm of the 3 numbers which add to 10..that ways
    Well then, you do want to solve is to find the number of partitions if the integer 10 into three or less summands.
    That is not a trivial problem. If you have access to a reasonably good mathematics library, the problem is discussed in Mathematics of Choice by Ivan Niven.
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