Find Range of Function

🎥 Video Explanation

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📝 Question

\[ f(x) = {}^{7-x}P_{x-3} \]

• A. {1, 2, 3, 4, 5}
• B. {1, 2, 3, 4, 5, 6}
• C. {1, 2, 3, 4}
• D. {1, 2, 3}

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✅ Solution

🔹 Step 1: Validity Conditions

For permutation \( ^nP_r \):

\[ n \ge r \ge 0 \]

Here:

\[ n = 7 – x, \quad r = x – 3 \]

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🔹 Step 2: Apply Conditions

\[ x – 3 \ge 0 \Rightarrow x \ge 3 \]

\[ 7 – x \ge x – 3 \Rightarrow 10 \ge 2x \Rightarrow x \le 5 \]

So: \[ x = 3, 4, 5 \]

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🔹 Step 3: Compute Values

x = 3:

\[ ^4P_0 = 1 \]

x = 4:

\[ ^3P_1 = 3 \]

x = 5:

\[ ^2P_2 = 2 \] —

🔹 Step 4: Range

\[ \{1, 2, 3\} \] —

🔹 Final Answer

\[ \boxed{\text{Option D}} \]