Question

The number of integral solutions of the equation $2x+2y+z=20$, where $x\ge 0,y\ge 0$ & $z\ge 0$ is

• A. $132$
• B. $11$
• C. $33$
• D. $66$

Answer 1

The correct option is D $66$

$2x+2y+z=20$

$\Rightarrow$ $2x+2y=20-z$

$\Rightarrow$ $x+y=10-\frac{z}{2}$

We have to find coefficient of ${\alpha }^{10-\frac{z}{2}}$ in $(1+\alpha +{\alpha }^2+{\alpha }^3+.....{)}^2$

Above is in G.P.

Coefficient of ${\alpha }^{10-\frac{z}{2}}$ in $(1-\alpha {)}^2$

Now,

Coefficient of ${\alpha }^{10-\frac{z}{2}}$ $={}^{11-\frac{z}{2}}{C}_{10-\frac{z}{2}}$ $=11-\frac{z}{2}$ ---- ( 1 )

$z$ is even and multiple of $2$

$z=0,2,4,6,8,10,12,14,16,18,20$

Substituting above all $z$ values in $\left(1\right)$ we get,

$\Rightarrow$ The number of integral solution $=11+10+9+8+7+6+5+4+3+2+1=66$