Amit M Agarwal Solutions for Chapter: Binomial Theorem, Exercise 6: Entrances Gallery

If ${}^{n}C_4,{}^{n}C_5$ and ${}^{n}C_6$ are in $\mathrm{AP}$, then $n$ is

${}^{15}C_3+{}^{15}C_5+\dots +{}^{15}C_{15}$ will be equal to

Let $C_0, C_1,\dots ,C_n$ denote the binomial coefficients in the expansion of ${\left(1+x\right)}^n.$ The value of $C_1-2C_2+3C_3-4C_4+\dots$ (upto $n$ terms) is

The expression ${}^{n}C_0+2·{}^{n}C_1+3·{}^{n}C_1+\dots +\left(n+1\right)·{}^{n}C_n$ is equal to

If in the expansion of ${\left(a-2b\right)}^n,$ the sum of the $5^{th}$ and $6^{th}$ terms is zero, then the value of $\frac{a}{b}$ is

The coefficient of the middle term in the expansion of ${\left(x+2y\right)}^6$ is

Let ${\left(1+x\right)}^n=1+a_{1}x+a_2{x}^2+\dots +a_n{x}^n.$ If $a_1, a_2$ and $a_3$ are in $\mathrm{AP}$, then the value of $n$ is

If the sum of the coefficients in the expansion of ${\left(a^2{x}^2-6ax+11\right)}^{10},$ where $a$ is constant is $1024,$ then the value of $a$ is