Question:
The co-efficient of $x^3$ in the expansion of $(1-x+x^2)^5$ is
The co-efficient of $x^3$ in the expansion of $(1-x+x^2)^5$ is
Updated On: Jul 7, 2022
- 10
- -20
- -50
- -30
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The Correct Option is D
Solution and Explanation
No. of terms in
$\left(1 - x+ x^{2}\right)^{5} = 1 + x\left(x- 1\right)^{5}$
$= \,^{5}C_{0} + \,^{5}C_{1}x\left(x-1\right)+ \,^{5}C_{2}x^{2}\left(x-1\right)^{2}$
$+ \,^{5}C_{3}x^{3} \left(x-1\right)^{3}+ \,^{5}C_{4}x^{4}\left(x-1\right)^{4}+ \,^{5}C_{5}x^{5}\left(x-1\right)^{5}$
$\therefore$ co - eff. $x^{3} = - 2\cdot^{5}C_{2}- \,^{5}C_{3} = -30$
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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 .