The range of the function $$f\left( x \right){ = ^{7 - x}}{P_{x - 3}}$$    is

Solution :
$$^{7 - x}{P_{x - 3}}$$   is defined if
$$\eqalign{ & 7 - x \geqslant 0,x - 3 \geqslant 0{\text{ and }}7 - x \geqslant x - 3 \cr & \Rightarrow 3 \leqslant x \leqslant 5\,{\text{and}}\,x \in I \cr & \therefore x = 3,4,5 \cr & \therefore f(3){ = ^{7 - 3}}{P_{3 - 3}}{ = ^4}{P_0} = 1 \cr & \therefore f(4){ = ^{7 - 4}}{P_{4 - 3}}{ = ^3}{P_1} = 3 \cr & \therefore f(5){ = ^{7 - 5}}{P_{5 - 3}}{ = ^2}{P_2} = 2 \cr} $$
Hence range = {I, 2, 3}