Question

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

Answer 1

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 $x_1+x_2+x_3+x_4=19.$