Binomial Theorem for Positive Integral Index

Sum of the series $\sum _{r=0}^{10}{\left(-1\right)}^r\left({}^{10}C_r\right)\left(\frac{1}{3^r}+\frac{8^r}{3^{2r}}\right)$ is

If $C_0, C_1, C_2, ..., C_n$ are the binomial coefficients in the expansion of ${\left(1+x\right)}^n$, then prove that $C_0-\frac{C_1}{\sqrt{2}}+\frac{C_2}{2}-\frac{C_3}{2\sqrt{2}}+....+\left(n+1\right) \mathrm{terms}$ equal to ${\left(1-\frac{1}{\sqrt{2}}\right)}^n$.

Let the coefficient of $x^{1009}$ in the polynomial equation $P\left(x\right)=\prod _{r=0}^{1009}\left(x+{}^{2019}C_r\right)$ is  in the form $m^n$ and sum of divisors of $n$ is the product of numbers $a$ and $b$ where, $a$ is composite number and $b$ is a three-digit prime number, then

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+...+C_n{x}^n$, then $\sum _{k=1}^n{k}^2{C}_k=$

Let $p=\sum _{r=1}^{2021}{\left(-1\right)}^{r-1}\frac{{}^{2021}C_r}{2^{r+1}-1}$ and $q=\sum _{r=1}^{\infty }\frac{{\left(2^r-1\right)}^{2021}}{2^{2022r}}$, then the value of $p+q$ is

The equation $a_8{x}^8+a_7{x}^7+a_6{x}^6+\dots +a_0=0$ has all roots positive and real and $a_8=1,a_7=-4$ and $a_0=\frac{1}{2^8}$, then the value of $\frac{a_2{a}_6}{a_4}$ is equal to ______

The remainder obtained when $(2109{)}^{360}-(1396{)}^{333}$ is divided by $7$ is

Values of $x$ for which the sixth term of the expansion of $E={\left(3^{{\log }_3\sqrt{9^{|x-2|}}}+7^{(1/5){\log }_7\left[\left(4\right)·3^{|x-2|}-9\right]}\right)}^7$ is $567$, are

Let $n$ digit number is formed using digits from the set $=\{2,3,4\}$ such that no two consecutive digits are $2$ . Let $a_n$ represents number of such numbers which ends with $2,b_n$ represents number of such numbers which ends with $3$ and $c_n$ represents number of such numbers which ends with $4$. Then which of the following is/are correct?

The value of $\sum _{r=0}^n\sum _{s=0}^n { }^n{C}_s.{ }^s{C}_r$ (where $r\le s$) is:

The larger of ${99}^{50}+{100}^{50}$ and ${101}^{50}$ is....

Sum of the last $30$ coefficients of powers of $x$ in the binomial expansion of ${\left(1+x\right)}^{59}$ is

If the digits at ten's and hundred's places in ${\left(11\right)}^{2016}$ are $x$ and $y$ respectively, then the ordered pair $\left(x,y\right)$ is equal to

For a positive integer $n$, if the mean of the binomial coefficients in the expansion of ${\left(a+b\right)}^{2n-3}$ is $16$, then $n$ is equal to

The sum of the series $S=\frac{1}{19!}+\frac{1}{3!·17!}+\frac{1}{5!·15!}+\dots \dots 10 \text{terms}$ is equal to

If $A$ and $B$ are the sum of the odd terms and even terms respectively in the expansion of ${\left(x+a\right)}^n$, then prove that $\left(A^2-B^2\right)={\left(x^2-a^2\right)}^n$ and $4AB={\left(x+a\right)}^{2n}-{\left(x-a\right)}^{2n}$.

If $a_1,a_2,a_3,a_4$ are coefficients of the four consecutive terms in the expansion of ${\left(1+x\right)}^n$, then prove that $\frac{a_1}{a_1+a_2}+\frac{a_3}{a_3+a_4}=\frac{2a_2}{a_2+a_3}$.

Show that $3^n-\frac{n}{1!}{3}^{n-1}+\frac{n\left(n-1\right)}{2!}{3}^{n-2}- ....+{(-1)}^n={}^{n}C_0+{}^{n}C_1+{}^{n}C_2+.... +{}^{n}C_n$.

Show that $2^m-\frac{m}{1!}{2}^{m-1}+\frac{m\left(m-1\right)}{2!}{2}^{m-2}- ....+{(-1)}^m=1$.

If the coefficients of the $2$nd, $3$rd, $4$th terms in the expansion of ${\left(1+x\right)}^{2n}$ are in the A.P. Show that $2n^2-9n+7=0$.