The number of integers$x, y, z, w$ such that $x + y + z + w = 20$and$x, y, z, w \ge – 1,$ is

Put $x=a-1,y=b-1,z=c-1,w=d-1$

then $a,b,c,d\ge 0\text{ and }(a-1)+(b-1)+(c-1)+(d-1)=20$

$\Rightarrow a+b+c+d=24$

The number of non-negative integral solutions of this equation
is

${ }^{24+4-1}{C}_{4-1}{=}^{27}{C}_3$