JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Permutations and Combinations

Here $a$ is even and $b$ is odd, so $a+b$ is odd and lies in $[3,199]$. The odd values with $a+b\equiv 2(\mathrm{mod}23)$ are $25,71,117,163$. Counting even $a$ for which $b=(a+b)-a$ stays in $\{1,3,\dots ,99\}$ gives $12$ pairs for $25$ (with $a=2,\dots ,24$), $35$ for $71$ ($a=2,\dots ,70$), $42$ for $117$ ($a=18,\dots ,100$), and $19$ for $163$ ($a=64,\dots ,100$). The total number of selections is $12+35+42+19=108$.