Question

The number of non-negative integral solutions of $3x+y+z=24$ is

• A. $117$
• B. $108$
• C. $225$
• D. $noneofthese$

Answer 1

The correct option is A $117$

Non-negative intergal solution at $3x+y+z=24$

$y+z=24-3x$

$x$ can be from $0$ to $8$

So, you have to basically find non-negative integral solutions of

$y+z=0,y+z=3,.....y+z=24$

It is basically distributing $n$ identical objects among $2$ groups such that each group have $o$ atg more $(\le n)$ objects.

Ways=$n{+}^{2-1}{C}_{2-1}{=}^{n+1}{C}_1$

Total solution${=}^1{C}_1{+}^4{C}_1{+}^7{C}_1+.....{.}^{25}{C}_1$

$=1+4+7+........25$

$=90+0+3+6+.....24$

$=90+3(1+2+.....8)$

$=90+\frac{3\times 8\times 9}{2}=9+108=117$

$A$ is correct.