The coefficient of the middle term in the expansion of $(2 + 3x)^4$ is :
- 6
- 5!
- 8!
- 216
The Correct Option is D
Solution and Explanation
So, total number of terms in $(2+3 x) 4=5$
Middle term is $3 rd . \Rightarrow T_{3}={ }^{4} C_{2}(2)^{2} \cdot(3 x)^{2}$
$=\frac{4 \times 3 \times 2 \times 1}{2 \times 1 \times 2} \times 4 \times 9 x^{2}$
$=216\, x^{2}$
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Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .