CombinatoricsSolutionIf I want to solve things with nonnegat

Combinatorics

Solution

If I want to solve things with non-negative values x1,x2,..,x4, I can simply consider some positive number y1 and then let x1=y1-1. That will produce all non-negative possibilities.

So you just solve:

x1 + x2 + x3 + x4 = 18
(y1 - 1) + (y2 - 1) + (y3 - 1) + (y4 - 1) = 18
y1 + y2 + y3 + y4 = 22

So now y1,y2,..,y4 are positive solutions -- we\'ve just \"transformed\" the problem into a problem we know the answer to, just with different parameters. The answer is 21C3 instead of 17C3.

We will use this trick repeatedly. To get the number of non-negative solutions to:
x1 + x2 + ... + xk = n
we will always do (x+k-1) choose (k-1)

And that is the answer to the question \"how many total?\" The answer is 21C3 =1330.