Equality of binomial coefficients (index transforms): Find all n such that C(35, n + 7) = C(35, 4n − 2).

Introduction / Context:Binomial symmetry says C(N, k) = C(N, N − k). Two binomial coefficients with the same top N are equal when their lower indices are equal or complementary with respect to N.

Given Data / Assumptions:N = 35; indices k1 = n + 7, k2 = 4n − 2.

Concept / Approach:Set up the two equality conditions: either k1 = k2 or k1 = 35 − k2. Solve both for integer n that keep indices within [0, 35].

Step-by-Step Solution:

Verification / Alternative check:Plugging back: C(35, 10) = C(35, 10) and C(35, 13) = C(35, 22) by symmetry — valid.

Why Other Options Are Wrong:Singletons 3 or 6 omit the second valid value; 28 is irrelevant to the solutions.

Common Pitfalls:Forgetting the complementary index possibility or neglecting domain limits.

Final Answer:3, 6