How many solutions are there to the equation u + v + w + x + y + z = 8, where u, v, w, x, y, and z are nonnegative integers, and x is at most 8?
+118609 There are 6 variables
One could be 8 and the rest zero 6 ways
One could be 7,1,0,0,0,0 6*5=30 ways
6,1,1,0,0,0 6*5C2 = 60 ways
6,2,0,0,0,0 6*5 = 30 ways
5,1,1,1,0,0 6*5C3 = 60 ways
5,2,1,0,0,0 6*5*4 = 120 ways
5,3,0,0,0,0 6*5 = 30 ways
4,1,1,1,1,0 6*5 = 30 ways
4,2,1,1,0,0 6*5*4C2 = 180 ways
4,2,2,0,0,0 6*5C2 = 60 ways
4,3,1,0,0,0 6*5*4 = 120 ways
4,4,0,0,0,0, 6*5 = 30 ways
3,1,1,1,1,1 6 ways
3,2,1,1,1,0 6*5*4 = 120 ways
3,2,2,1,0,0 6*5*4C2 = 180 ways
3,3,1,1,0,0 6C2*4C2 = 90 ways
3,3,2,0,0,0 6C2*4 = 60 ways
2,2,2,2,1,1 6C2=15 ways
2,2,2,2,2,0 6 ways
Add all those up and I think you will have your answer. :)
