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 .............. .
Largest coefficient in the expansion of $(1+x)^{30}={ }^{30} \mathrm{C}_{30 / 2}={ }^{30} \mathrm{C}_{15}$
The number of terms in the expansion of $(x+y+z)^n$ ............ .
Given expansion is $(x+y+z)^n=[x+(y+z)]^n$.
$$\begin{aligned} {[x+(y+z)]^n={ }^n C_0 x^n+{ }^n C_1 x^{n-1}(} & (y) z) \\ & +{ }^n C_2 x^{n-2}(y+z)^2+\ldots+{ }^n C_n(y+z)^n \end{aligned}$$
$$\begin{aligned} \therefore \text { Number of terms } & =1+2+3+\ldots+n+(n+1) \\ & =\frac{(n+1)(n+2)}{2} \end{aligned}$$
In the expansion of $\left(x^2-\frac{1}{x^2}\right)^{16}$, the value of constant term is ............. .
Let constant be $T_{t+1}$.
$$\begin{aligned} T_{r+1} & ={ }^{16} C_r\left(x^2\right)^{16-r}\left(-\frac{1}{x^2}\right)^r \\ & ={ }^{16} C_r x^{32-2 r}(-1)^r x^{-2 r} \\ & ={ }^{16} C_r x^{32-4 r}(-1)^r \end{aligned}$$
For constant term, $\quad 32-4 r=0 \Rightarrow r=8$
$$\therefore \quad T_{8+1}={ }^{16} C_8$$