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

The correct option is B $224$

$x+y+z=15-5t,t=0,1,2,3.$
The number of non-negative integer solutions is
${}^{15-5t+3-1}{C}_{3-1}={}^{17-5t}{C}_2=\frac{(17-5t)(16-5t)}{2}$
The desired number is $\sum _{t=0}^3\frac{(17-5t)(16-5t)}{2}=136+66+21+1=224$