Question

Let n C_r denote the binomial coefficient of x^r in the expansion of (1 + x)^n. If ∑_{k=0}^{10} (2² + 3k) n C_k = α 3^{10} + β 2^{10}, α, β ∈ R, then α + β is equal to ________.

Hint:

Use the identity $\sum_{k=0}^n k \binom{n}{k} = n 2^{n-1}$ for quick calculations of binomial sums involving $k$.

Answer 1

Correct Answer: 19

Step 1: Break the sum (assuming $n=10$): $\sum_{k=0}^{10} 4 \binom{10}{k} + \sum_{k=0}^{10} 3k \binom{10}{k}$.
Step 2: First part: $4 \sum \binom{10}{k} = 4 \cdot 2^{10}$.
Step 3: Second part: $3 \sum k \binom{10}{k} = 3 \sum k \frac{10}{k} \binom{9}{k-1} = 30 \sum_{k=1}^{10} \binom{9}{k-1} = 30 \cdot 2^9 = 15 \cdot 2^{10}$.
Step 4: Total $= (4 + 15) 2^{10} = 19 \cdot 2^{10}$.
Step 5: Comparing with $\alpha 3^{10} + \beta 2^{10}$: $\alpha = 0, \beta = 19$. $\alpha + \beta = 19$.