Find the range

We have 3 Restrictions :

1) $7-x\ge0 \implies x\le7$

2) $x-3\ge0 \implies x\ge3$

3) $7-x\ge x-3 \implies x\le5$

Taking Intersection of 1,2 and 3 , we get $3\le x\le5$

since $x\in I$ domain of $^{7-x}P_{x-3}$ is { ${3,4,5}$ } , so Range of the Function is { ${^{4}P_{0},^{3}P_{1},^{2}P_{2}}$ }

Answer $\displaystyle= ^{4}P_{0}+ ^{3}P_{1}+ ^{2}P_{2} = \color{#3D99F6}{\boxed{6}}$