Question:
The coefficient of $x^2$ in the expansion of the product $(2-x^2)??(1+2x+3x^2)^6 + (1-4x^2)^6)$ is :
The coefficient of $x^2$ in the expansion of the product $(2-x^2)??(1+2x+3x^2)^6 + (1-4x^2)^6)$ is :
Updated On: Jul 29, 2022
- 107
- 106
- 108
- 155
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The Correct Option is B
Solution and Explanation
coefficient of $x^2= 2$ coefficient of $x^2$ in $((1 + 2x + 3x^2)^6 + (1 - 4x^2)6)$ - constant term
$(1 + 2x + 3x^2)6 \sum_\limits{r=0}^6$
$^{6}C_{r}\left(2x+3x^{2}\right)^{r}$
$=^{6}C_{0}+^{6}C_{1}\left(2x+3x^{2}\right)+^{6}C_{2} \left(2x+3x^{2}\right)^{2}+ ...$
coefficient of $x^2 = 2(18 + 60 - 24) - 2$
$= 108 - 2 = 106$
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Notes on binomial expansion formula
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 .