JEE Main · 2004 · 24 Apr 2004 · Offline · Permutations and Combinations

The symbol ${}^{7-x}{P}_{x-3}$ only makes sense when the lower index is a non-negative integer not exceeding the upper index, so I need $x-3\ge 0$ and $7-x\ge x-3$. The first gives $x\ge 3$ and the second gives $2x\le 10$, that is $x\le 5$, so $x$ can only be $3,4$ or $5$. Evaluating at these, $f(3)={}^4{P}_0=1$, $f(4)={}^3{P}_1=3$ and $f(5)={}^2{P}_2=2$, so the function takes exactly the values $\{1,2,3\}$.