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 ${}^{17-5t}{C}_2=\frac{\left(17-5t\right)\left(16-5t\right)}{2}$
Hence, the desired number is equal to
$\sum _{t=0}^3\frac{\left(17-5t\right)\left(16-5t\right)}{2}$
$=\frac{17\times 16}{2}+\frac{12\times 11}{2}+\frac{7\times 6}{2}+\frac{2\times 1}{2}$
$=136+66+21+1$
$=224$