Sum of the coefficients of x^4 and x^6 in the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The expansion is given by $(1 + (x - x^2))^6$. Using the binomial theorem,$(1 + y)^n = \sum_{k=0}^{n} \binom{n}{k} y^k$. Here,$n = 6$ and $y = (x

Vedclass · Accepted Answer

$11$

Vedclass · Answer

(C) The expansion is given by $(1 + (x - x^2))^6$. Using the binomial theorem,$(1 + y)^n = \sum_{k=0}^{n} \binom{n}{k} y^k$. Here,$n = 6$ and $y = (x - x^2)$. $(1 + x - x^2)^6 = \sum_{k=0}^{6} \binom{6}{k} (x - x^2)^k = \sum_{k=0}^{6} \binom{6}{k} x^k (1 - x)^k$. To find the coefficient of $x^4$: For $k=2$: $\binom{6}{2} x^2 (1 - x)^2 = 15 x^2 (1 - 2x + x^2) = 15x^2 - 30x^3 + 15x^4$. Coefficient is $15$. For $k=3$: $\binom{6}{3} x^3 (1 - x)^3 = 20 x^3 (1 - 3x + 3x^2 - x^3) = 20x^3 - 60x^4 + 60x^5 - 20x^6$. Coefficient is $-60$. For $k=4$: $\binom{6}{4} x^4 (1 - x)^4 = 15 x^4 (1 - 4x + 6x^2 - 4x^3 + x^4) = 15x^4 - 60x^5 + 90x^6 - 60x^7 + 15x^8$. Coefficient is $15$. Total coefficient of $x^4 = 15 - 60 + 15 = -30$. To find the coefficient of $x^6$: For $k=3$: $\binom{6}{3} x^3 (1 - x)^3 = 20 x^3 (1 - 3x + 3x^2 - x^3) = 20x^3 - 60x^4 + 60x^5 - 20x^6$. Coefficient is $-20$. For $k=4$: $\binom{6}{4} x^4 (1 - x)^4 = 15 x^4 (1 - 4x + 6x^2 - 4x^3 + x^4) = 15x^4 - 60x^5 + 90x^6 - 60x^7 + 15x^8$. Coefficient is $90$. For $k=5$: $\binom{6}{5} x^5 (1 - x)^5 = 6 x^5 (1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5) = 6x^5 - 30x^6 + 60x^7 - 60x^8 + 30x^9 - 6x^{10}$. Coefficient is $-30$. For $k=6$: $\binom{6}{6} x^6 (1 - x)^6 = 1 x^6 (1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6) = x^6 - 6x^7 + 15x^8 - 20x^9 + 15x^{10} - 6x^{11} + x^{12}$. Coefficient is $1$. Total coefficient of $x^6 = -20 + 90 - 30 + 1 = 41$. Sum of coefficients $= -30 + 41 = 11$.

Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $(1+x-x^2)^6$ is

Similar Questions

If the coefficients of $x^2$ and $x^3$ in the expansion of $(3 + ax)^9$ are the same,then the value of $a$ is

If the sum of the coefficients in the expansion of $(x+y)^{n}$ is $4096,$ then the greatest coefficient in the expansion is .... .

If the $2^{\text{nd}}$,$3^{\text{rd}}$,and $4^{\text{th}}$ terms in the expansion of $(x+a)^{n}$ are $96, 216$,and $216$ respectively,and $n$ is a positive integer,then $a+x=$

The term independent of $x(x>0, x \neq 1)$ in the expansion of $\left[\frac{(x+1)}{\left(x^{2 / 3}-x^{1 / 3}+1\right)}-\frac{(x-1)}{(x-\sqrt{x})}\right]^{10}$ is:

If $k$ is the coefficient of $x^5$ in the expansion of $(2x^2 - \frac{1}{3x^3})^5$,then $\frac{3k}{2} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module