How many solutions are there to the equation u + v + w + x + y + z = 2, where $u,$ $v,$ $w,$ $x,$ $y,$ and $z$ are nonnegative integers, and $x$ is at most $1?$
Case 1
x = 0 and one of the other variables = 2 → 5 different solutions
Case 2
x = 0 and two of the other variables = 1 → C(5,2) = 10 different solutions
Case 3
x = 1 and one of the other variables = 1 → C(5,1) = 5 different solutions
5 + 10 + 5 = 20 possible solutions
+529 
