Binomial Theorem

A set S contains (2n +1) elements. There are 4096 subset of S which contain at most n elements. what is n equal to

What is $C(n, 1)+C(n, 2)+\ldots+C(n, n)$ equal to?

What is the coefficient of the middle term in the expansion of $\left(1+4 x+4 x^2\right)^5$ ?

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: Consider the following statements :…

If the constant term in the expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is 405 , then what can be the values of $k$ ?

If $k < (\sqrt{2}+1)^3 < k+2$, where $k$ is a natural number, then what is the value of $k$?

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

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

How many terms are there in the expansion of $\left(\frac{a^2}{b^2}+\frac{b^2}{a^2}+2\right)^{21}$ where $a \neq 0, b \neq 0$ ?

How many terms are there in the expansion of $(3 x-y)^4(x+3 y)^4$ ?

If the sixth term in the binomial expansion of $\left(x^{-\frac{8}{3}}+x^2 \log _{10} x\right)^8$ is 5600 , then what is the value of $x$ ?

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