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Question
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Options
{1, 2, 3, 4, 5}
{1, 2, 3, 4, 5, 6}
{1, 2, 3, 4}
{1, 2, 3}
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Solution
We know that
\[7 - x > 0; x - 3 \geq 0 \text{ and }7 - x \geq x - 3\]
\[ \Rightarrow x < 7; x \geq 3 \text{ and }2x \leq 10\]
\[ \Rightarrow x < 7; x \geq 3 \text{ and }x \leq 5\]
\[So,x = \left\{ 3, 4, 5 \right\}\]
\[\text{ Range of }f\]
\[=\left\{\ {^ \left( 7 - 3 \right)}{}{P}_\left( 3 - 3 \right) , \ {^\left( 7 - 4 \right)}{}{P}_\left( 4 - 3 \right) , {^\left( 7 - 5 \right)}{}{P} \left( {}_{5 - 3} \right) \right\}\]
\[=\left\{ 4 P_0 , 3 P_1 , 2 P_2 \right\}\]
\[=\left\{ 1, 3, 2 \right\}\]
\[=\left\{ 1, 2, 3 \right\}\]
So, the answer is (d).
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