Question

The number of non-negative integral solutions of the equation $x+y+z+5t=15$ is

• A. $196$
• B. $224$
• C. $312$
• D. $364$

Answer 1

Answer: (B)

$x+y+z+5t=15,t\in \left\{0 ,1 , 2 ,3\right\}$
$\Rightarrow x+y+z=15-5t$
Number of non-negative integral solutions of
$x+y+z=15-5t$ is equal to $^{17 - 5 t}C_{2}=\frac{\left(17 - 5 t\right) \left(16 - 5 t\right)}{2}$
Hence, the desired number is equal to
$\displaystyle \sum _{t = 0}^{3} \frac{\left(17 - 5 t\right) \left(16 - 5 t\right)}{2}$
$=\frac{17 \times 16}{2}+\frac{12 \times 11}{2}+\frac{7 \times 6}{2}+\frac{2 \times 1}{2}$
$=136+66+21+1$
$=224$