JEE Main 2023 (Online) 11th April Morning ShiftQuestion 1
The number of triplets $(x,y,z)$, where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15$, is
Permutation and Combination
Solution
Non-negative solutions of $x+y+z=15$: ${}^{17}C_2=136$. Solutions with $x=y$: $z=15-2x\ge0$, $x=0..7$, so 8; same for each of 3 pairs. Solutions with all equal: $x=y=z=5$, 1. Solutions with some equality $=3\cdot8-3\cdot1+1=22$. Distinct triplets $=136-22=114$. Option B.