Question:
The coefficient of $x^4$ in the expansion of $(1 + x + x^2 + x^3)^{11}$, is
The coefficient of $x^4$ in the expansion of $(1 + x + x^2 + x^3)^{11}$, is
Updated On: Apr 1, 2024
- 440
- 770
- 990
- 1001
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The Correct Option is C
Solution and Explanation
We have coefficient of $x^4$ in $(1 + x + x^2 + x^3)^{11}$
= coefficient of $x^4$ in $(1 + x^2)^{11} (1 + x)^{11}$
= coefficient of $x^4$ in $(1 + x)^{11}$ + coefficient of $x^2$ in
$ 11. (1 + x)^{11}$ + constant term is $^{11}C_2. (1 + x)^{11}$
$ = {^{11}C_4} + 11 . {^{11}C_2} + {^{11}C_2} = 990 $
= coefficient of $x^4$ in $(1 + x^2)^{11} (1 + x)^{11}$
= coefficient of $x^4$ in $(1 + x)^{11}$ + coefficient of $x^2$ in
$ 11. (1 + x)^{11}$ + constant term is $^{11}C_2. (1 + x)^{11}$
$ = {^{11}C_4} + 11 . {^{11}C_2} + {^{11}C_2} = 990 $
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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 .