Question

Given: \[ \sum_{k=0}^{5} \binom{10}{2k} = \alpha \quad \text{and} \quad \sum_{k=0}^{4} \binom{10}{2k+1} = \beta \] \text{Find the value of} \( \alpha - \beta \).

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

In problems involving binomial coefficients, use symmetry to simplify the calculations. The sum of even and odd binomial coefficients in the expansion of \( (1 + 1)^{10} \) are equal.

Answer 1

The Correct Option is A

We are given the following sums: \[ \sum_{k=0}^{5} \binom{10}{2k} = \alpha \quad \text{and} \quad \sum_{k=0}^{4} \binom{10}{2k+1} = \beta \] We need to find the value of \( \alpha - \beta \).

1. Step 1: Use the binomial expansion: The binomial expansion of \( (1 + 1)^{10} \) is: \[ (1 + 1)^{10} = \sum_{k=0}^{10} \binom{10}{k} \] This simplifies to: \[ 2^{10} = \sum_{k=0}^{10} \binom{10}{k} \]

2. Step 2: Split the sum into even and odd terms: The sum of the even-indexed binomial coefficients is: \[ \sum_{k=0}^{5} \binom{10}{2k} = \alpha \] and the sum of the odd-indexed binomial coefficients is: \[ \sum_{k=0}^{4} \binom{10}{2k+1} = \beta \] By symmetry of binomial coefficients, we know that: \[ \alpha = \beta \]

3. Step 3: Conclusion: Since \( \alpha = \beta \), it follows that: \[ \alpha - \beta = 0 \] Thus, the correct answer is \( 0 \), corresponding to option (A).