Embibe Experts Solutions for Chapter: Binomial Theorem, Exercise 2: Level 2

If in the binomial expansion of $(1-x{)}^m(1+x{)}^n$, the coefficients of $x$ and $x^2$ are respectively $3$ and $-4$, then the ratio $m: n$ is equal to

Find the last two digits of the greatest integer less than ${\left(2+\sqrt{3}\right)}^4.$

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+\dots +(1+x{)}^{30}$ is

The coefficient of $x^4$ in the expansion of ${\left(1+x+x^2+x^3\right)}^{11}$ is

If $a_i\left(i=0, 1, 2,\dots , 16\right)$ be real constants such that for every real value of $x$, ${\left(1+x+x^2\right)}^8=a_0+a_{1}x+a_2{x}^2+\dots +a_{16}{x}^{16}$, then $a_5$ is equal to

The sum $\frac{1}{1!2004!}+\frac{1}{3!2002!}+\frac{1}{5!2000!}+\dots +\frac{1}{2003!2!}=\left(\frac{2^a-1}{ b!}\right)$ where $a$ and $b$ are positive integers. Then
find $b-a.$

If in the expansion of ${\left(1+x\right)}^n; C_0, C_1, C_2, \dots C_n$ denote the binomial coefficients, then the value of $\frac{C_o}{1}+\frac{C_2}{3}+\frac{C_4}{5}+\frac{C_6}{7}+\dots$ 

In the expansion of ${\left(1+\frac{3x}{2}\right)}^{-5}$ the coefficient of $x^{10}$ is equal to the coefficient of $x^{10}$ in ${\left(1+ax\right)}^n,n\in N$ then $na=$