Binomial Theorem for Positive Integral Index

It is given that the value in $n{C}_r$, $r>n$ if $r$ is positive integer

By using binomial expansion we get that the sum of last two digits of $3^{100}$ is equal to

By using binomial expansion we get that the sum of last three digits of $3^{100}$ is not equal to $1$

It is given that if ${25}^{15}$ is divided by $13,$ then we find a remainder as $P$. Find $P$

By using binomial theorem, we know that $6^n-5n-1$ is always divisible by $25,$ where $n$ is a

Expand ${\left(x-a\right)}^5$

Expand ${\left(u-v\right)}^5$

Write the number of terms (only numerical value) in the expansion of ${\left(a^2+y\right)}^{10}$.

Write the number of terms (only numerical value) in the expansion of ${\left(x^2+y\right)}^{20}$.

Find the fourth term from the end in the expansion of ${\left(x-\frac{1}{x}\right)}^{10}$

Find the fourth term from the end in the expansion of ${\left(2x-\frac{1}{x^2}\right)}^{10}$

The value of ${}^{n}C_r$ is $P$, where $n=8$ and $r=2$, then the value of $P$ is

The value of ${}^{n}C_r$ is $P$, where $n=12$ and $r=4$, then the value of $P$ is

Expand ${\left(\frac{x}{5}+\frac{4}{y}\right)}^4$ by using Binomial theorem

Expand ${\left(\frac{x}{3}+\frac{2}{y}\right)}^4$ by using Binomial theorem

The value of ${}^{15}C_8={}^{15}C_9$

The value of ${}^{11}C_5=$ _____.

${}^{25}{C}_{10}+{}^{25}{C}_9=$

${}^n{C}_0$ _____ ${}^n{C}_n$

In the expansion of ${\left(1+B\right)}^{m+n}$, prove that coefficient of $B^m$ and $B^n$ are equal