Binomial Theorem for Positive Integral Index

What is the binomial expansion of ${\left(1+x\right)}^4?$

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 $2n^2-9n+7=0$.

Find the value of $n$ so that the coefficients of the $2$nd, $3$rd and $4$th terms in the expansion of ${(1+x)}^n$ are in A.P.

Show that ${\left(\sqrt{2}+1\right)}^6+{\left(\sqrt{2}-1\right)}^6=198$.

Find the value of ${\left(x-\sqrt{1-x^2}\right)}^4+{\left(x+\sqrt{1-x^2}\right)}^4$.

Find the term independent of $x$ in $\left(1+2x+x^3\right){\left(x-\frac{1}{x}\right)}^5$

Find the coefficient of $x$ in $\left(1-2x^2+3x^5\right){\left(1+\frac{1}{x}\right)}^{10}$

Find the coefficient of $x$ in $\left(1-x^2+2x^4\right){\left(1-\frac{1}{x}\right)}^6$

Find the coefficient of $x^r$ in $\left(1+2x-x^2\right){\left(1+x\right)}^n, 0<r<n+2$

Find the coefficient of $x^r$ in $\left(1-x\right){\left(1+2x\right)}^m, where 0<r<m+1$

Find the coefficient of $x^3$ in the expansion of $\left(2+3x-4x^2\right){\left(1-x\right)}^8$

Write down the binomial expansion of the following :

${\left({\mathrm{a}}^{\frac{2}{3}}+{\mathrm{b}}^{\frac{1}{2}}\right)}^4$

Write down the binomial expansion of the following :

${\left(x+y-2\right)}^4$

Write down the binomial expansion of the following :

${\left(x-\frac{1}{x}\right)}^6$

Write down the binomial expansion of the following :

${\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^7$

Write down the binomial expansion of the following :

${\left(a+b\right)}^5$