Question

Suppose A and B are the coefficients of the 30th and 12th terms respectively in the binomial expansion of \( (1 + x)^{2n - 1} \). If \( 2A = 5B \), then \( n \) is equal to:

• A. \( 20 \)
• B. \( 22 \)
• C. \( 21 \)
• D. \( 19 \)

Hint:

When working with binomial expansions, use the general term formula \( T_k = \binom{2n-1}{k} x^k \) to extract the coefficients and find relationships between terms.

Answer 1

The Correct Option is C

In the binomial expansion of \( (1 + x)^{2n - 1} \), the general term is given by: \[ T_k = \binom{2n-1}{k} x^k. \] The 30th term corresponds to \( T_{30} \), and the 12th term corresponds to \( T_{12} \). We are given that \( 2A = 5B \), where \( A \) and \( B \) are the coefficients of the 30th and 12th terms respectively. Solving the equation \( 2A = 5B \), we can find the value of \( n \). 
Final Answer: \( n = 21 \).