Question

If \( A \) and \( B \) are binomial coefficients of the 30\(^\text{th}\) and 12\(^\text{th}\) terms of the binomial expansion \( (1 + x)^{2n-1} \), and \( 2A = 5B \), then the value of \( n \) is

• A. 20
• B. 21
• C. 14
• D. 20

Hint:

When solving binomial coefficient problems, use the properties of binomial expansions to express terms and simplify the equations. Equating coefficients can help solve for unknowns like \( n \).

Answer 1

The Correct Option is B

The binomial expansion of \( (1 + x)^{2n-1} \) is given by the general term: \[ T_{r+1} = \binom{2n-1}{r} x^r \] For the 30\(^\text{th}\) term, we use: \[ T_{30} = \binom{2n-1}{29} = A \] For the 12\(^\text{th}\) term, we use: \[ T_{12} = \binom{2n-1}{11} = B \] We are given that: \[ 2A = 5B \] Substituting the expressions for \( A \) and \( B \): \[ 2 \binom{2n-1}{29} = 5 \binom{2n-1}{11} \] By solving this equation, we find that \( n = 21 \).