General Term and Middle Term in a Binomial Expansion

The number of rational terms in the expansion of ${\left(7^{1/3}+{11}^{1/9}\right)}^{6561}$ is given as $P$. Find the value of $P$.

Given that the number of rational terms in the expansion of ${\left(9^{1/4}+8^{1/6}\right)}^{1000}$ is less than $500$

Given that greatest binomial coefficient containing $x^8$ in the expansion of $(x^2-y{)}^{10}$. Find the coefficient

It is given that the coefficient of $x^{10}$ in the expansion of $(x^2-2{)}^{11}$ is the greatest coefficient. Find the coefficient.

Find the middle term of ${\left(x-\frac{1}{x}\right)}^5$ by using binomial expansion.

Find the middle term of ${\left(3x-\frac{1}{2x}\right)}^9$ by using binomial expansion.

The middle term in the expansion of ${\left(a-b\right)}^6$ is $N{a}^3{b}^3$, where $N$ is

The middle term in the expansion of ${\left(x-\frac{1}{y}\right)}^n$ is independent of $n$ when $n=4$

The number of terms $T$ in the expansion of ${\left(a+b\right)}^m$ is $T<m$, when $m$ is

The number of terms in the expansion of ${\left(a+b\right)}^{50}$ is one less than $50$

Find $B$ if the coefficients of $y^2$ and $y^3$ in the expansion of ${\left(3+By\right)}^9$ are equal.

Find $x, y$ and $z$ in the expansion of ${\left(x+y\right)}^z$ if the first three terms of the expansion are $729, 7290 \text{and} 30375$, respectively.

Prove that the coefficient of $x^n$ in the expansion of ${\left(1+x\right)}^{2n}$ is twice the coefficient of $x^n$ in the expansion of ${\left(1+x\right)}^{2n-1}$.

The coefficients of the ${\left(r-1\right)}^{th}, r^{th}$ and ${\left(r+1\right)}^{th}$ terms in the expansion of ${\left(x+1\right)}^n$ are in the ratio $1:3:5.$ Find $n$ and $r.$

The coefficient of $a^5{b}^7$ in ${\left(a-2b\right)}^{12}$ is $101376$.

Find the coefficient of $x^3$ in the expansion of ${\left(x+3\right)}^8$

Find the coefficient of $x^6{y}^3$ in the expansion of ${\left(x+2y\right)}^9$.

Find the middle terms in the expansion of ${\left(5x-7y\right)}^7$

The middle term in the expansion of ${\left(2x+3y\right)}^8$ is ${}^{8}C_4{\left(x\right)}^4{\left(y\right)}^4$

Find the $3^{\text{rd}}$ term in the expansion of ${\left(3x-5y\right)}^7$.