Question:
The constant term in the expansion of $\left(x^2 - \frac{1}{x^2} \right)^{16}$ is
The constant term in the expansion of $\left(x^2 - \frac{1}{x^2} \right)^{16}$ is
Updated On: Oct 25, 2024
- ${^{16}C_{8}}$
- ${^{16}C_{7}}$
- ${^{16}C_{9}}$
- ${^{16}C_{10}}$
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The Correct Option is A
Solution and Explanation
General term, $T_{r+1} = {^{16}C_r} (x^2)^{16-r} \left( - \frac{1}{x^2} \right)^r = {^{16}C_r}(-1)^r x^{32-4r}$
For constant term, $32 - 4r = 0$
$\Rightarrow$ r = 8
$\therefore$ Constant term, $T_9 = {^{16}C_{8}}$
For constant term, $32 - 4r = 0$
$\Rightarrow$ r = 8
$\therefore$ Constant term, $T_9 = {^{16}C_{8}}$
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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 .