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Given (56)P(r+6) : (54)P(r+3) = 30800 : 1, find the integer value of r. (Use nPr = n! / (n − r)! and simplify the ratio carefully to isolate r.)

Difficulty: Medium

40
41
42
None of these

Correct Answer: 41

Explanation:

Introduction / Context:
This question tests algebraic manipulation with permutations. Ratios of nPr expressions often telescope when written in factorial form, letting us solve for the unknown r quickly.

Given Data / Assumptions:

(56)P(r+6) : (54)P(r+3) = 30800 : 1.
nPr = n! / (n − r)!, for valid integer ranges.
We assume r is an integer making all factorial terms defined.

Concept / Approach:
Rewrite each permutation in factorial form and simplify the ratio. Many terms cancel, leaving a small product in r. Then equate to the given ratio to solve for r.

Step-by-Step Solution:

(56)P(r+6) = 56! / (56 − (r+6))! = 56! / (50 − r)!(54)P(r+3) = 54! / (54 − (r+3))! = 54! / (51 − r)!Ratio = [56!/(50 − r)!] / [54!/(51 − r)!] = (56!/54!) * [(51 − r)!/(50 − r)!]= (56 * 55) * (51 − r) = 3080 * (51 − r)Given 3080 * (51 − r) = 30800 ⇒ 51 − r = 10 ⇒ r = 41.

Verification / Alternative check:
Plug r = 41 back into the simplified expression: 3080 * (51 − 41) = 3080 * 10 = 30800, which matches the given ratio exactly.

Why Other Options Are Wrong:
40 and 42 would give 3080 * 11 or 3080 * 9, not 30800. “None of these” is unnecessary because a listed option is correct.

Common Pitfalls:
Misplacing (n − r)! when expanding nPr, or forgetting that only adjacent factorial factors survive after cancellation.

Given (56)P(r+6) : (54)P(r+3) = 30800 : 1, find the integer value of r. (Use nPr = n! / (n − r)! and simplify the ratio carefully to isolate r.)

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