Question:
If \(^{10}P_r = 604800\) and \(^{10}C_r = 120\), then \(r\) is:
If \(^{10}P_r = 604800\) and \(^{10}C_r = 120\), then \(r\) is:
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Remember that \(nP_r = nC_r \cdot r!\); use this relation to find unknown \(r\) when \(nP_r\) and \(nC_r\) are given.
Updated On: Jul 18, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Recall formulas.
\[ ^nP_r = \frac{n!}{(n-r)!}, \quad ^nC_r = \frac{n!}{r!(n-r)!} \]
Step 2: Express factorial relation.
\[ ^nP_r = ^nC_r \cdot r! \Rightarrow 604800 = 120 \cdot r! \Rightarrow r! = \frac{604800}{120} = 5040 \]
Step 3: Solve for \(r\).
\[ 7! = 5040 \Rightarrow r = 7 \]
Step 4: Final conclusion.
\[ \boxed{7} \]
\[ ^nP_r = \frac{n!}{(n-r)!}, \quad ^nC_r = \frac{n!}{r!(n-r)!} \]
Step 2: Express factorial relation.
\[ ^nP_r = ^nC_r \cdot r! \Rightarrow 604800 = 120 \cdot r! \Rightarrow r! = \frac{604800}{120} = 5040 \]
Step 3: Solve for \(r\).
\[ 7! = 5040 \Rightarrow r = 7 \]
Step 4: Final conclusion.
\[ \boxed{7} \]
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