Question

If the sum of the coefficients in the expansion of \( (x + y)^n \) is 1024, then the value of the greatest coefficient in the expansion is:

• A. 356
• B. 252
• C. 210
• D. 120

Hint:

In a binomial expansion \( (x + y)^n \), the sum of the coefficients is \( 2^n \), and the greatest coefficient typically occurs at \( k = \frac{n}{2} \) for even \( n \).

Answer 1

The Correct Option is B

We are given that the sum of the coefficients in the expansion of \( (x + y)^n \) is 1024. 
The sum of the coefficients in the expansion of \( (x + y)^n \) is given by: \[ \text{Sum of the coefficients} = (1 + 1)^n = 2^n \] We are told that the sum of the coefficients is 1024, so: \[ 2^n = 1024 \] Solving for \( n \): \[ n = \log_2{1024} = 10 \] Thus, the expansion is for \( (x + y)^{10} \). 
Find the greatest coefficient 
The general term in the binomial expansion of \( (x + y)^{10} \) is: \[ T_k = \binom{10}{k} x^{10-k} y^k \] The coefficient of the term \( T_k \) is \( \binom{10}{k} \). The greatest coefficient in the expansion occurs at \( k = 5 \), as the binomial coefficients are symmetric and maximized at the middle term for even powers. Thus, the greatest coefficient is: \[ \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \] Therefore, the greatest coefficient in the expansion is 252, which corresponds to Option B.