Question

Find the number of non-negative integral solutions of $3x+y+z=24$.

Answer 1

We have
$3x+y+z=24$ and given $x\ge 0,y\ge 0,z\ge 0$
Let $x=k$
$\therefore y+z=24-3k...\left(i\right)$
Here $24\ge 24-3k\ge 0(\because x\ge 0)$
Hence, $0\le k\le 8$
The total number of integral solutions of $\left(i\right)$ is
${}^{24-3k+2-1}{C}_{2-1}{=}^{25-3k}{C}_1=25-3k$
Hence the total number of solutions of the original equation
$=\sum _{k=0}^8(25-3k)=25\sum _{k=0}^{8}1-3\sum _{k=0}^{8}k$
$=25.9-3\frac{8.9}{2}=225-108=117$