Suppose that a nonnegative integer solution to the equation u + v + w + x + y = 12 How many solutions exist where u = 1

Question

I'm fairly new to combinatorics and I'm stumped by this problem.

So far I've figured out the total number of solutions...

C(16,4) = 1820 possible solutions

But I'm not sure what the next step is to find the solutions where u = 1.

Thanks for the help.

Answer 1

Guide:

Setting $u=1$, the problem become

$$v+w+x+y = 11$$

Now, you can repeat your trick to solve the similar problem where $v,w,x,y$ are nonnegative.

The answer can again be expressed in the form of $\binom{m}{k}$.