If C(50, r) = C(50, r+2), find r. (Use symmetry C(n, k) = C(n, n − k) and solve for k.)

Introduction / Context:Equal binomial coefficients at positions r and r+2 imply a reflection around the center via C(n, k) = C(n, n − k). We use this to relate r and r+2.

Given Data / Assumptions:

• C(50, r) = C(50, r+2).
• 0 ≤ r, r+2 ≤ 50 (implicit domain).

Concept / Approach:Nontrivial equality happens when r = 50 − (r+2). Solve for r.

Step-by-Step Solution:

Verification / Alternative check:Check: C(50, 24) = C(50, 26) by symmetry, consistent with the condition.

Why Other Options Are Wrong:23, 22, 21 do not satisfy r = 50 − (r+2).

Common Pitfalls:Assuming equality implies r = r+2 (impossible) instead of using symmetry across n/2.

Final Answer:24