Question

Coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \)

• A. 100
• B. 50
• C. 25
• D. 75

Hint:

Use the binomial theorem to expand binomials, and find the powers of terms by identifying how the powers of \( x \) are derived.

Answer 1

The Correct Option is B

We need to find the coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \). This is a binomial expansion of the form \( (a + b)^n \), where: \[ a = 4, \quad b = -\frac{x^2}{9}, \quad n = 12 \] The general term in the binomial expansion is given by: \[ T_k = \binom{n}{k} a^{n-k} b^k \] Substituting the values for \( a \), \( b \), and \( n \): \[ T_k = \binom{12}{k} 4^{12-k} \left( -\frac{x^2}{9} \right)^k \] We want the power of \( x^9 \), so: \[ 2k = 9 \quad \Rightarrow \quad k = 4.5 \] Since \( k \) must be an integer, the coefficient of \( x^9 \) does not exist as a whole number term in the expansion.