Binomial Theorem

The value of $\displaystyle\sum_{r=1}^{15}r^2\left(\frac{{}^{15}C_r}{{}^{15}C_{r-1}}\right)$ is equal to :

If the coefficients of $x^{-2}$ and $x^{-4}$ in the expansion of $\left(x^{1/3}+\dfrac1{2x^{1/3}}\right)^{18}$, (x > 0), are m and n respectively, then $\dfrac mn$ is equal to :

The sum of coefficients of integral powers of $x$ in the binomial expansion of $\left(1-2\sqrt x\right)^{50}$ is :

If the coefficients of the three successive terms in the binomial expansion of $(1+x)^n$ are in the ratio 1 : 7 : 42, then the first of these terms in the expansion is :

Let X be a set containing 10 elements and P(X) be its power set. If A and B are picked up at random from P(X), with replacement, then the probability that A and B have equal number of elements, is :

The term independent of $x$ in the binomial expansion of
$$\left(1-\frac1x+3x^5\right)\left(2x^2-\frac1x\right)^8$$
is :

Coefficient of $x^{11}$ in the expansion of $(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$ is

If $X=\{4^n-3n-1:n\in N\}$ and $Y=\{9(n-1):n\in N\}$, where N is the set of natural numbers, then $X\cup Y$ is equal to :

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1+ax+bx^2)(1-2x)^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to :

The number of terms in the expansion of $(1+x)^{101}\left(1+x^2-x\right)^{100}$ in powers of $x$ is :

The coefficient of $x^{50}$ in the binomial expansion of
$$(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\dots+x^{1000}$$
is :

If $\left(2+\dfrac x3\right)^{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are :

If $1+x^4+x^5=\displaystyle\sum_{i=0}^5a_i(1+x)^i$, for all $x$ in $\mathbb R$, then $a_2$ is :

The coefficient of $x^{1012}$ in the expansion of $(1+x^n+x^{253})^{10}$, (where $n\le22$ is any positive integer), is :

The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio 5 : 10 : 14. Then $n=$

For $r=0,1,\ldots,10$, let $A_r$, $B_r$ and $C_r$ denote, respectively, the coefficient of $x^r$ in the expansions of $(1+x)^{10}$, $(1+x)^{20}$ and $(1+x)^{30}$. Then $$\sum_{r=1}^{10}A_r\,(B_{10}B_r-C_{10}A_r)$$ is equal to

Let
$$L=\lim_{x\to0}\frac{a-\sqrt{a^2-x^2}-\dfrac{x^2}4}{x^4},\qquad a>0.$$
If $L$ is finite, then