Question

The coefficient of \( x^{14}y \) in the expansion of \( (x^2 + \sqrt{y})^9 \) is:

• A. 84
• B. 36
• C. 63
• D. 252
• E. 128

Hint:

When finding a specific term in a binomial expansion, identify the powers required for each component of the term (e.g., \(x\) and \(y\)). Use the binomial coefficient formula to calculate the coefficient for the term by matching these powers with the general term expression in the binomial theorem. This method allows precise and efficient calculation of any specific term in the expansion.

Answer 1

The Correct Option is B

To find the coefficient of \( x^{14}y \) in the binomial expansion of \( (x^2 + \sqrt{y})^9 \), consider the general term in the binomial expansion, which is given by: \[ T_k = \binom{9}{k} (x^2)^{9-k} (\sqrt{y})^k. \] We want the term where the power of \( x \) is 14 and the power of \( y \) is 1. Since \( x \) appears in the term \( (x^2)^{9-k} \), we set: \[ 2(9-k) = 14 \quad \Rightarrow \quad 18 - 2k = 14 \quad \Rightarrow \quad 2k = 4 \quad \Rightarrow \quad k = 2. \] Plugging \( k = 2 \) into the term for \( y \): \[ (\sqrt{y})^2 = y^1. \] This is the correct power of \( y \), and now we compute the coefficient: \[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36. \] 
Conclusion: The coefficient of \( x^{14}y \) in the expansion is 36, corresponding to option (B).