If C(n+2, 8) : P(n−2, 4) = 57 : 16, find the integer value of n. (Use C(a, b) = a! / (b! (a − b)!) and P(a, b) = a! / (a − b)!)

Introduction / Context:
We must simplify a ratio mixing a combination and a permutation. Converting each to factorials and canceling common factors typically yields a manageable polynomial equation in n.

Given Data / Assumptions:

• C(n+2, 8) / P(n−2, 4) = 57/16.
• n is an integer large enough that all factorials are defined (n ≥ 19 from the solution).

Concept / Approach:
Write C(n+2, 8) = (n+2)! / (8! (n−6)!) and P(n−2, 4) = (n−2)! / (n−6)!. Then the ratio collapses to ((n+2)! / 8!) / (n−2)! = (n+2)(n+1)n(n−1) / 8!.

Step-by-Step Solution:

Verification / Alternative check:
Nearby n = 18 gives 116280; n = 20 gives 20212219 = larger than target, confirming n = 19 uniquely fits.

Why Other Options Are Wrong:
17, 18, 20 do not satisfy the exact product 143640.

Common Pitfalls:
Forgetting that the (n−6)! terms cancel; mis-evaluating 8! or arithmetic with large products.

Final Answer:
19