Question:
If ${}^{n}P_{r}=840$ and ${}^{n}C_{r}=35$, then the value of $r$ is equal to
If ${}^{n}P_{r}=840$ and ${}^{n}C_{r}=35$, then the value of $r$ is equal to
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Combinatorics Tip: Memorize the first few factorials ($3!=6$, $4!=24$, $5!=120$, $6!=720$) to make solving these relationship equations instantly recognizable!
Updated On: Apr 30, 2026
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The Correct Option is B
Solution and Explanation
Concept:
Permutations (${}^nP_r$) and Combinations (${}^nC_r$) are mathematically linked. A permutation accounts for the arrangement of selected items, while a combination only accounts for their selection. The relationship formula is: $${}^nP_r = r! \times {}^nC_r$$
Step 1: Identify the given values.
From the problem statement: $${}^nP_r = 840$$ $${}^nC_r = 35$$
Step 2: Substitute into the relationship formula.
Use the identity connecting permutations and combinations: $$840 = r! \times 35$$
Step 3: Isolate the factorial term.
Divide both sides of the equation by 35 to solve for $r!$: $$r! = \frac{840}{35}$$
Step 4: Calculate the value of r!
Perform the division: $$r! = 24$$
Step 5: Determine the integer value of r.
We must find an integer $r$ such that $r \times (r-1) \times \dots \times 1 = 24$. Testing factorials: $1! = 1$ $2! = 2$ $3! = 6$ $4! = 4 \times 3 \times 2 \times 1 = 24$ Therefore, $r = 4$. Hence the correct answer is (B) 4.
Permutations (${}^nP_r$) and Combinations (${}^nC_r$) are mathematically linked. A permutation accounts for the arrangement of selected items, while a combination only accounts for their selection. The relationship formula is: $${}^nP_r = r! \times {}^nC_r$$
Step 1: Identify the given values.
From the problem statement: $${}^nP_r = 840$$ $${}^nC_r = 35$$
Step 2: Substitute into the relationship formula.
Use the identity connecting permutations and combinations: $$840 = r! \times 35$$
Step 3: Isolate the factorial term.
Divide both sides of the equation by 35 to solve for $r!$: $$r! = \frac{840}{35}$$
Step 4: Calculate the value of r!
Perform the division: $$r! = 24$$
Step 5: Determine the integer value of r.
We must find an integer $r$ such that $r \times (r-1) \times \dots \times 1 = 24$. Testing factorials: $1! = 1$ $2! = 2$ $3! = 6$ $4! = 4 \times 3 \times 2 \times 1 = 24$ Therefore, $r = 4$. Hence the correct answer is (B) 4.
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