Question:
The co-efficient of $x^{-12}$ in the expansion of$\left(x+\frac{y}{x^3}\right)^{20}$is
The co-efficient of $x^{-12}$ in the expansion of$\left(x+\frac{y}{x^3}\right)^{20}$is
Updated On: Jul 7, 2022
- $\,^{20}C_8$
- $\,^{20}C_8y^8$
- $\,^{20}C_{12}$
- $\,^{20}C_{12}\,^y12$
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The Correct Option is B
Solution and Explanation
Suppose $x^{-12}$ occurs is (r + 1)th term.
We have
$T_{r+1} = ^{20} C_{r} x^{20-r} \left(\frac{y}{x^{3}}\right)^{r}$
$ = ^{20}C_{r} x^{20 -4r} y^{r}$
This term contains $ x^{-12}$ if $ 20-4r=-12$ or $ r = 8 $ .
$\therefore $ The coefficient of $x^{-12}$ is ${^{20}C_{8}} \, y^{8} $
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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 .