The coefficient of t^{12} in the expansion of | vedclass

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

Question

(C) The given expression is $(1 + t^2)^6(1 + t^6)(1 + t^{12})$.Expanding the last two terms: $(1 + t^6)(1 + t^{12}) = 1 + t^6 + t^{12} + t^{18}$.So,th

Vedclass · Accepted Answer

$22$

Vedclass · Answer

(C) The given expression is $(1 + t^2)^6(1 + t^6)(1 + t^{12})$.Expanding the last two terms: $(1 + t^6)(1 + t^{12}) = 1 + t^6 + t^{12} + t^{18}$.So,the expression becomes $(1 + t^2)^6(1 + t^6 + t^{12} + t^{18})$.We need the coefficient of $t^{12}$ in the product $(1 + t^2)^6(1 + t^6 + t^{12} + t^{18})$.This is equal to:(Coefficient of $t^{12}$ in $(1 + t^2)^6 	imes 1$) + (Coefficient of $t^6$ in $(1 + t^2)^6 	imes t^6$) + (Coefficient of $t^0$ in $(1 + t^2)^6 	imes t^{12}$).Using the binomial expansion $(1 + t^2)^6 = \sum_{k=0}^{6} {^6C_k} (t^2)^k = \sum_{k=0}^{6} {^6C_k} t^{2k}$.$1$. Coefficient of $t^{12}$ in $(1 + t^2)^6$ is ${^6C_6} = 1$.$2$. Coefficient of $t^6$ in $(1 + t^2)^6$ is ${^6C_3} = \frac{6 	imes 5 	imes 4}{3 	imes 2 	imes 1} = 20$.$3$. Coefficient of $t^0$ in $(1 + t^2)^6$ is ${^6C_0} = 1$.Summing these: $1 + 20 + 1 = 22$.

The coefficient of $t^{12}$ in the expansion of $(1 + t^2)^6(1 + t^6)(1 + t^{12})$ is:

Similar Questions

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 number of rational terms in the binomial expansion of ${\left( 7^{\frac{1}{7}} + 11^{\frac{1}{11}} \right)^{711}}$ is:

If $f(n)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$,then $f(2021)=$

If $p$ and $q$ are respectively the coefficients of $x^{-3}$ and $x^{-5}$ in the expansion of $\left(x^{1/3} + \frac{1}{2x^{1/3}}\right)^{21}, x > 0$,then $\frac{5p}{4q} = $

The number of integral terms in the expansion of $(\sqrt{3} + \sqrt[8]{5})^{256}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module