Question:
The coefficient of $ x^{2} $ in the expansion of $ (1 + x + x^{2} + x^{3})^{10} $ is
The coefficient of $ x^{2} $ in the expansion of $ (1 + x + x^{2} + x^{3})^{10} $ is
Updated On: Jun 23, 2024
- $ 42 $
- $ 43 $
- $ 44 $
- $ 55 $
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The Correct Option is D
Solution and Explanation
The given expansion is,
$(1 + x + x^2 + x^3)10 = [1 + x + x^2(1 + x)]^{10}$
$= [(1 + x)(1 + x^2)]^{10} = (1 + x )^{10}(1 + x^2)^{10}$
$=(1 + \,^{10}C_1x + \,^{10}C_2x^2 + ....+ \,^{10}C_{10}x^{10})$
$(1 + \,^{10}C_1x^2 + \,^{10}C_2x^4 + ....+ \,^{10}C_{10} x^{20})$
$\therefore$ Coefficient of $x^2 = \,^{10}C_1 + \,^{10}C_2 = 55$
$(1 + x + x^2 + x^3)10 = [1 + x + x^2(1 + x)]^{10}$
$= [(1 + x)(1 + x^2)]^{10} = (1 + x )^{10}(1 + x^2)^{10}$
$=(1 + \,^{10}C_1x + \,^{10}C_2x^2 + ....+ \,^{10}C_{10}x^{10})$
$(1 + \,^{10}C_1x^2 + \,^{10}C_2x^4 + ....+ \,^{10}C_{10} x^{20})$
$\therefore$ Coefficient of $x^2 = \,^{10}C_1 + \,^{10}C_2 = 55$
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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 .