Question

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

• A. $116$
• B. $117$
• C. $118$
• D. $119$

Answer 1

The correct option is B $117$

Given $3x+y+z=24$

If $x=0$ , then $y+z=24$ , so possible integral solutions are $25$

If $x=1$ , then $y+z=21$ , so possible integral solutions are $22$

If $x=2$ , then $y+z=18$ , so possible integral solutions are $19$

If $x=3$ , then $y+z=15$ , so possible integral solutions are $16$

If $x=4$ , then $y+z=12$ , so possible integral solutions are $13$

If $x=5$ , then $y+z=9$ , so possible integral solutions are $10$

If $x=6$ , then $y+z=6$ , so possible integral solutions are $7$

If $x=7$ , then $y+z=3$ , so possible integral solutions are $4$

If $x=8$ , then $y+z=0$ , so possible integral solutions are $1$

The total number of possible integral solutions is $25+22+19+16+13+10+7+4+1=117$