Question:
If ${}^9P_5 = (504)({}^6P_r)$, then the value of $r$ is equal to:
If ${}^9P_5 = (504)({}^6P_r)$, then the value of $r$ is equal to:
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Always simplify factorial expressions step-by-step instead of expanding fully.
Updated On: Apr 24, 2026
- $3$
- $2$
- $1$
- $4$
- $5$
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The Correct Option is B
Solution and Explanation
Concept:
• ${}^nP_r = \dfrac{n!}{(n-r)!}$
Step 1: Evaluate ${}^9P_5$
\[ {}^9P_5 = \frac{9!}{4!} = 9 \times 8 \times 7 \times 6 \times 5 = 15120 \]
Step 2: Substitute in equation
\[ 15120 = 504 \cdot {}^6P_r \]
Step 3: Solve for ${}^6P_r$
\[ {}^6P_r = \frac{15120}{504} = 30 \]
Step 4: Find $r$
\[ {}^6P_2 = 6 \times 5 = 30 \] Final Conclusion:
\[ r = 2 \]
• ${}^nP_r = \dfrac{n!}{(n-r)!}$
Step 1: Evaluate ${}^9P_5$
\[ {}^9P_5 = \frac{9!}{4!} = 9 \times 8 \times 7 \times 6 \times 5 = 15120 \]
Step 2: Substitute in equation
\[ 15120 = 504 \cdot {}^6P_r \]
Step 3: Solve for ${}^6P_r$
\[ {}^6P_r = \frac{15120}{504} = 30 \]
Step 4: Find $r$
\[ {}^6P_2 = 6 \times 5 = 30 \] Final Conclusion:
\[ r = 2 \]
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