The coefficient of x^{n-6} in the expansion n | vedclass

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

Question

(A) The general term in the product is of the form $\left[ \frac{x}{k} - \frac{^nC_{k-1} + ^nC_k}{^nC_{k-1}} \right]$.Using the identity $\frac{^nC_k}

Vedclass · Accepted Answer

$^nC_6 (n+1)^6$

Vedclass · Answer

(A) The general term in the product is of the form $\left[ \frac{x}{k} - \frac{^nC_{k-1} + ^nC_k}{^nC_{k-1}} \right]$.Using the identity $\frac{^nC_k}{^nC_{k-1}} = \frac{n-k+1}{k}$,we have $\frac{^nC_{k-1} + ^nC_k}{^nC_{k-1}} = 1 + \frac{n-k+1}{k} = \frac{k+n-k+1}{k} = \frac{n+1}{k}$.Thus,the expression becomes $n! \left( x - (n+1) \right) \left( \frac{x}{2} - \frac{n+1}{2} \right) \dots \left( \frac{x}{n} - \frac{n+1}{n} \right)$.Factoring out $\frac{1}{k}$ from each bracket,we get $n! \cdot \frac{1}{1 \cdot 2 \cdot 3 \dots n} (x - (n+1))^n$.Since $n! = 1 \cdot 2 \cdot 3 \dots n$,the expression simplifies to $(x - (n+1))^n$.The coefficient of $x^{n-6}$ in $(x - (n+1))^n$ is given by the binomial theorem as $^nC_6 (x)^{n-6} (-(n+1))^6 = ^nC_6 (n+1)^6 x^{n-6}$.Therefore,the coefficient is $^nC_6 (n+1)^6$.

Similar Questions

The number of integral terms in the expansion of $\{7^{(1/2)} + 11^{(1/6)}\}^{824}$ is equal to ...................

If $A$ and $B$ are the coefficients of $x^n$ in the expansions of $(1 + x)^{2n}$ and $(1 + x)^{2n - 1}$ respectively,then

In the expansion of $(1+x)^n$,the coefficients of the $p^{th}$ and $(p+1)^{th}$ terms are respectively $p$ and $q$. Then $p+q$ is equal to:

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}$, where $\alpha \neq 0$, are equal, then the value of $\alpha$ is:

Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of $(2x^{1/5} - x^{-1/5})^{15}$,$x > 0$,be $m$ and $n$ respectively. If $r$ is a positive integer such that $mn^2 = {}^{15}C_r \cdot 2^r$,then the value of $r$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module