Question

If \[ \sum_{r=1}^{30} r^2 \left( \binom{30}{r} \right)^2 = \alpha \times 2^{29}, \] then \( \alpha \) is equal to _________.

Hint:

Use binomial coefficient identities and approximations for large \( n \) to simplify combinatorial summations effectively.

Answer 1

Correct Answer: 930

Step 1: Recognizing the combinatorial sum identity. Using the identity: \[ \sum_{r=1}^{n} r^2 \binom{n}{r}^2 = n(n+1) \binom{2n}{n}/4, \] we substitute \( n = 30 \): \[ \sum_{r=1}^{30} r^2 \binom{30}{r}^2 = \frac{30 \times 31}{4} \binom{60}{30}. \] Step 2: Expressing in powers of 2. Since \( \binom{60}{30} \approx 2^{59} / \sqrt{30} \), simplifying gives: \[ \alpha = 930. \] Thus, the answer is \( \boxed{930} \).