How many solutions in non-negative integers are there to the equation x1 + x2+ x3 + x4 = 19

Theorem

Let $n$ and $k$ are positive integers. Then the number of non-negative integer solutions of the equation $x_1+x_2+...+x_n=k$ is given by $C(k+n-1, k).$

Given $n=4, k=19.$

Then the number of non-negative integer solutions of the equation

$x_1+x_2+x_3+x_4=19$

is

$C(19+4-1, 19)=\dbinom{22}{19}=\dfrac{22!}{3!(22-19)!}$

$=\dfrac{22(21)(20)}{1(2)(3)}=1540$

There are 1540 non-negative integer solutions of the equation $x_1+x_2+x_3+x_4=19.$