Binomial Theorem

How many terms are there in the expansion of $\left(1+\frac{2}{x}\right)^9\left(1-\frac{2}{x}\right)^9$ ?

What is the coefficient of the middle term in the binomial expansion of $(2+3 \mathrm{x})^4$ ?

Let $(1+x)^n=1+T_1 x+T_2 x^2+T_3 x^3+\ldots+T_n x^n$ What is $T_1+2 T_2+3 T_3+\ldots+n T_n$ equal to?

Consider the following statements for a fixed natural number $n$ : $\mathrm{C}(n, r)$ is greatest if $n=2 r$ $\mathrm{C}(n, r)$ is greatest if $n=2 r-1$ and $n=2 r+1$ Which of the statements given above is/are correct?

Let $(1+x)^n=1+T_1 x+T_2 x^2+T_3 x^3+\ldots+T_n x^n$ What is $1-T_1+2 T_2-3 T_3+\ldots+(-1)^n n T_n$ equal to?

What is the remainder when $5^{99}$ is divided by $13$?

What is the coefficient of $x^{10}$ in the expansion of $\left(1-x^2\right)^{20}\left(2-x^2-\frac{1}{x^2}\right)^{-5}$ ?

Passage: Let $u$ be a positive integer and $f$ be a real number lying between $0$ and $1$. Further, $\left(\sqrt{2}+1\right)^{10}=u+f$ and $\left(\sqrt{2}-1\right)^{10}=v$. Question: What is the multiplicative inverse…

What is the number of rational terms in the expansion of $\left(3^{\frac{1}{2}} + 5^{\frac{1}{4}}\right)^{12}$?

How many terms are there in the expansion of $\left(1+2 x+x^2\right)^5+\left(1+4 y+4 y^2\right)^5$ ?

In a binomial expansion of $(x+y)^{2 n+1}(x-y)^{2 n+1}$, the sum of middle terms is zero. What is the value of $\left(\frac{x^2}{y^2}\right)$ ?

What is the number of terms in the expansion of $\left[(2 x-3 y)^2\right.$ $\left.(2 \mathrm{x}+3 \mathrm{y})^2\right]^2 ?$