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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2018 · Mathematics · Binomial Theorem

What is the coefficient of the middle term in the binomial...

What is the coefficient of the middle term in the binomial expansion of (2+3x)4(2+3 \mathrm{x})^4 ?

Options

  1. A6
  2. B12
  3. C108
  4. D216

Correct answer

D. 216

Step-by-step solution

Middle term in the expansion of (x+y)n=(n+12)th (x+y)^n=\left(\frac{n+1}{2}\right)^{\text{th }} term, if nn is odd =(n2+1)th =\left(\frac{n}{2}+1\right)^{\text{th }} term, if nn is even Here n=4∴n=4\therefore Middle term is \left(\frac{4}{2}+1\right)^{\text{th }}=3^{\text{rd }}termterm4c2×22×32=6×4×9=216\begin{aligned} 4 c_2 \times 2^2 \times 3^2 & =6 \times 4 \times 9 \\ & =216 \end{aligned}$
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