Consider the equation a+b+c+d = 12, where a b c d are non-negative integers. We're looking for the number of solutions this equation has. That is to say, we are looking for number of ways we can distribute 12 among the 4 variables a, b, c, and d.

We have 12+3=15 places (12 for each 1 adding up to 12, and 3 for the 3 plus signs to combine a b c d together), where we put stars (for numbers) and bars (for plus signs), one item per place. The key idea is that this configuration stands for a solution to our equation. For example

{*|*****|****|**}

Stands for the solution 1+5+4+2=12. Similarly

{|*****|***|****}

Denotes the solution 0+5+3+4=12, because we have no star at first, then a bar, and similar reasoning like the previous.

We see that any such configuration stands for a solution to the equation, and any solution to the equation can be converted to such a stars-bars series. So our problem reduces to "in how many ways can we place stars and bars in places?" This is the same as fixing places out of places and filling the rest with stars. We can do this in, of course, C(3, 15) ways (combination, not permutation, of choosing 3 out of 15). So the number of solutions to our equation is C(3, 15) = 455.