JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Permutations and Combinations

Total non-negative integer solutions to $x+y+z=15$ is $\left(\genfrac{}{}{0ex}{}{17}{2}\right)=136$. We subtract cases where at least two variables are equal. When exactly two are equal (say $x=y\ne z$): $2t+z=15$ gives 7 solutions (excluding $t=5$ where all three are equal). By symmetry, there are $3\times 7=21$ such solutions. When all three are equal: $x=y=z=5$ gives 1 solution. Required answer = $136-21-1=114$.