Find the term independent of $x$, where $x \neq 0$, in the expansion of $\left(\frac{3 x^2}{2}-\frac{1}{3 x}\right)^{15}$.

If the term free from $x$ in the expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is 405 , then find the value of $k$.

Find the coefficient of $x$ in the expansion of $\left(1-3 x+7 x^2\right)(1-x)^{16}$.

Find the term independent of $x$ in the expansion of $\left(3 x-\frac{2}{x^2}\right)^{15}$.

Find the middle term (terms) in the expansion of

(i) $\left(\frac{x}{a}-\frac{a}{x}\right)^{10}$

(ii) $\left(3 x-\frac{x^3}{6}\right)^9$

Find the coefficient of $x^{15}$ in the expansion of $\left(x-x^2\right)^{10}$.

Find the coefficient of $\frac{1}{x^{17}}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$.

Find the sixth term of the expansion $\left(y^{1 / 2}+x^{1 / 3}\right)^n$, if the Binomial coefficient of the third term from the end is 45.

Find the value of $r$, if the coefficients of $(2 r+4)$ th and $(r-2)$ th terms in the expansion of $(1+x)^{18}$ are equal.

If the coefficient of second, third and fourth terms in the expansion of $(1+x)^{2n}$ are in A.P, then show that $2n^2-9n+7=0$.

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

If $p$ is a real number and the middle term in the expansion of $\left(\frac{p}{2}+2\right)^8$ is 1120 , then find the value of $p$.

Show that the middle term in the expansion of $\left(x-\frac{1}{x}\right)^{2 n}$ is $$\frac{1 \times 3 \times 5 \times \ldots \times(2 n-1)}{n!} \times(-2)^n$$

Find $n$ in the Binomial $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n$, if the ratio of 7 th term from the beginning to the 7 th term from the end is $\frac{1}{6}$.

In the expansion of $(x+a)^n$, if the sum of odd terms is denoted by 0 and the sum of even term by $E$. Then, prove that

(i) $O^2-E^2=\left(x^2-a^2\right)^n$.

(ii) $40 E=(x+a)^{2 n}-(x-a)^{2 n}$.

If $x^p$ occurs in the expansion of $\left(x^2+\frac{1}{x}\right)^{2 n}$, then prove that its coefficient is $\frac{2 n!}{\frac{(4 n-p)!}{3!} \frac{(2 n+p)!}{3!}}$.

Find the term independent of $x$ in the expansion of

$$\left(1+x+2 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$$

18 The total number of terms in the expansion of $(x+a)^{100}+(x-a)^{100}$ after simplification is

If the integers $r>1, n>2$ and coefficients of $(3 r)$ th and $(r+2)$ nd terms in the Binomial expansion of $(1+x)^{2 n}$ are equal, then

The two successive terms in the expansion of $(1+x)^{24}$ whose coefficients are in the ratio $1:4$ are

The coefficient of $x^n$ in the expansion of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ are in the ratio

If the coefficients of 2 nd , 3 rd and the 4 th terms in the expansion of $(1+x)^n$ are in AP, then the value of $n$ is

If $A$ and $B$ are coefficient of $x^n$ in the expansions of $(1+x)^{2 n}$ and $(1+x)^{2 n-1}$ respectively, then $\frac{A}{B}$ equals to

If the middle term of $\left(\frac{1}{x}+x \sin x\right)^{10}$ is equal to $7 \frac{7}{8}$, then the value of $x$ is

The largest coefficient in the expansion of $(1+x)^{30}$ is .............. .

The number of terms in the expansion of $(x+y+z)^n$ ............ .

In the expansion of $\left(x^2-\frac{1}{x^2}\right)^{16}$, the value of constant term is ............. .

If the seventh term from the beginning and the end in the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n$ are equal, then $n$ equals to ............ .

The coefficient of $a^{-6} b^4$ in the expansion of $\left(\frac{1}{a}-\frac{2 b}{3}\right)^{10}$ is $\qquad$

Middle term in the expansion of $\left(a^3+b a\right)^{28}$ is ............ .

The ratio of the coefficients of $x^p$ and $x^q$ in the expansion of $(1+x)^{p+q}$ is .............. .

The position of the term independent of $x$ in the expansion of $\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^2}\right)^{10}$ is ............. .

If $25^{15}$ is divided by 13 , then the remainder is .............. .

The sum of the series $\sum_\limits{r=0}^{10}{ }^{20} C_r$ is $2^{19}+\frac{{ }^{20} C_{10}}{2}$.

The expression $7^9+9^7$ is divisible by 64 .

The number of terms in the expansion of $\left[\left(2 x+y^3\right)^4\right]^7$ is 8 .

The sum of coefficients of the two middle terms in the expansion of $(1+x)^{2 n-1}$ is equal to ${ }^{2 n-1} C_n$.

The last two digits of the numbers $3^{400}$ are 01 .

If the expansion of $\left(x-\frac{1}{x^2}\right)^{2 n}$ contains a term independent of $x$, then $n$ is a multiple of 2.

The number of terms in the expansion of $(a+b)^n$, where $n \in N$, is one less than the power $n$.