The coefficient of $x^{10}$ in the expansion of $1+\left(1+x\right)+\ldots+\left(1+x\right)^{20}$ is
- $^{19}C_{9}$
- $^{20}C_{10}$
- $^{21}C_{11}$
- $^{22}C_{12}$
The Correct Option is C
Solution and Explanation
The correct option is(C): \(^{21}C_{11}\)
The given series is in GP. Hence, its sum
\(S=\frac{1\left\{(1+x)^{20+1}-1\right\}}{(1+x)-1}=\frac{(1+x)^{21}-1}{x}\)
Therefore, the required coefficient of $x^{10}$ in the expansion of \(\frac{(1+x)^{21}-1}{x}\)
$=$ Coefficient of $x^{11}$ in the expansion of \((1+x)^{21}-1\)
\(={ }^{21} C_{11}\)
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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 .