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Number Theory 4D Space

  1. Apr 3, 2012 #1
    1. The problem statement, all variables and given/known data

    How many points (x1,x2,x3,x4) in the 4-dimensional space with nonnegative integer coordinates satisfy the equation x1 + x2 + x3 + x4 = 10?

    I'm not sure which method to use to start this problem. Any ideas?
     
  2. jcsd
  3. Apr 3, 2012 #2

    jedishrfu

    Staff: Mentor

    forget the 4D for the moment and consider all the ways you can add up numbers to get 10:

    0+0+0+10 = 10
    0+0+1+9 = 10
    1+1+1+7 = 10
    ...

    and next you consider for each 4 number sum and ask yourself how many 4D points can have coordinate points of 1,1,1,7:
    (1,1,1,7) (1,1,7,1) (1,7,1,1) (7,1,1,1)
     
  4. Apr 3, 2012 #3
    That will take forever to use brute force.

    Thinking combinatorially, my initial thought would be C(10+4-1,3) = C(13,3) = 286 different ways. Any thoughts?
     
  5. Apr 4, 2012 #4

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    That's correct.

    More generally, can you show that the number of solutions to ##x_1 + x_2 + \cdots + x_k = n## with each x_i a nonnegative integer is C(n+k-1,k-1)?
     
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