JEE Main 2022 (Online) 25th July Evening ShiftQuestion 1
The number of bijective functions $f:\{1,3,5,7,\ldots,99\}\to\{2,4,6,8,\ldots,100\}$, such that $f(3)\ge f(9)\ge f(15)\ge f(21)\ge\ldots\ge f(99)$, is
Permutation and Combination
Solution
The 17 inputs $3,9,\ldots,99$ must map to 17 distinct values in strictly decreasing order, so choose the 17-element set of images: $\binom{50}{17}$ ways, with a unique order. The other 33 inputs map to the remaining 33 values in $33!$ ways. Total $=\binom{50}{17}33!=\frac{50!}{17!}={}^{50}P_{33}$. Option (B).