JEE 2023 — Mathematics PYQ

First, total number of non-negative integer solutions of  
$x+y+z=15$ is

$\left(\genfrac{}{}{0ex}{}{15+3-1}{3-1}\right)=\left(\genfrac{}{}{0ex}{}{17}{2}\right)=136$

Now subtract the cases where at least two variables are equal.

Case I: $x=y$

Then $2x+z=15\Rightarrow z=15-2x$

Possible non-negative $x$ values: $x=0,1,2,3,4,5,6,7$

$\Rightarrow 8$ solutions

Similarly for $y=z$ and $x=z$

Total $=3\times 8=24$

But cases where $x=y=z$ are counted thrice.

$x=y=z\Rightarrow 3x=15\Rightarrow x=5$

So, one such solution.

Hence, solutions with at least two equal variables:

$24-2=22$

Therefore, required number of distinct solutions:

$136-22=114$