Question

The coefficient of the middle term in the expansion of \( (2 + 3x)^4 \) is

• A. 216
• B. 200
• C. 180
• D. 2160

Hint:

In a binomial expansion, the middle term is found by selecting the appropriate value of \( k \) based on the degree of the expansion.

Answer 1

The Correct Option is D

Step 1: Use the binomial expansion formula.
The expansion of \( (2 + 3x)^4 \) is: \[ (2 + 3x)^4 = \sum_{k=0}^{4} \binom{4}{k} 2^{4-k} (3x)^k \]
Step 2: Find the middle term.
The middle term occurs when \( k = 2 \). Thus, the middle term is: \[ \binom{4}{2} 2^{4-2} (3x)^2 = 6 \times 4 \times 9x^2 = 216x^2 \]
Step 3: Conclusion.
The coefficient of the middle term is 216. Final Answer: \[ \boxed{216} \]