Question

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

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

Answer 1

Answer: (B)

$224$

If $t=0:x+y+z=15$. This has ${}^{15+3-1}{C}_{3-1}=136$ solutions.
If $t=1:x+y+z=10$. This has ${}^{10+3-1}{C}_{3-1}=66$ solutions.
If $t=2:x+y+z=5$. This has ${}^{5+3-1}{C}_{3-1}=21$ solutions.
If $t=3:x+y+z=0$. This has only $1$ solution.
Thus, answer $=136+66+21+1=224$
Hence, (B) is correct.