Find the number of integral solutions of the equation x + y + z = 20 where x, y, z > 0.

• When we are to find the total number of ways to distribute k indistinguishable objects among n distinct entities such that each entity receives at least one object.
  
  x1+x2+x3+⋯+xn =k ∀ x1,x2,x3,⋯,xn ≥ 1
  
  The number of ways is given by k−1Cn−1.
  
  For this case, We can say that x,y and z cannot be more than 18. Start with z=1,2,3,,...,18
  For z=1, x+y=19=>18 solutions.
  For z=2,x+y=18=>17 solutions
  Forz=3,x+y=17=>16 dolutions
  .................................................
  For z=18,x+y=2=>1 solution
  Hence sum of all solutions is
  =18+17+16+...............+1
  =(18*19)/2=9*19=171
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