Question

How many possible combinations of the variables a,b,c,d,e are possible if I know that:

a+b+c+d+e = 500 

and that they are all integers and >= 0, so I know they are finite.

Answer 1

@Torlack, @Jason Cohen: Recursion is a bad idea here, because there are ‘overlapping subproblems.’ I.e., If you choose a as 1 and b as 2, then you have 3 variables left that should add up to 497; you arrive at the same subproblem by choosing a as 2 and b as 1. (The number of such coincidences explodes as the numbers grow.)

The traditional way to attack such a problem is dynamic programming: build a table bottom-up of the solutions to the sub-problems (starting with ‘how many combinations of 1 variable add up to 0?’) then building up through iteration (the solution to ‘how many combinations of n variables add up to k?’ is the sum of the solutions to ‘how many combinations of n-1 variables add up to j?’ with 0 <= j <= k).

public static long getCombos( int n, int sum ) {   // tab[i][j] is how many combinations of (i+1) vars add up to j   long[][] tab = new long[n][sum+1];   // # of combos of 1 var for any sum is 1   for( int j=0; j < tab[0].length; ++j ) {     tab[0][j] = 1;   }   for( int i=1; i < tab.length; ++i ) {     for( int j=0; j < tab[i].length; ++j ) {       // # combos of (i+1) vars adding up to j is the sum of the #       // of combos of i vars adding up to k, for all 0 <= k <= j       // (choosing i vars forces the choice of the (i+1)st).       tab[i][j] = 0;       for( int k=0; k <= j; ++k ) {         tab[i][j] += tab[i-1][k];       }     }   }   return tab[n-1][sum]; } 

 $ time java Combos 2656615626  real    0m0.151s user    0m0.120s sys 0m0.012s