Binomial Theore...

The first three terms in the expansion of $$(1+a)^n$$ are $$t_1=1,t_2=-18,t_3=144$$.

Use the general term to determine a and n.

The value of $$a_0 ^2-a_1 ^2+a_2 ^2....a_{2n} ^2$$ is

The middle term of expansion of $$\left (\dfrac {10}{x} + \dfrac {x}{10}\right )^{10}$$

The value of $$C_1 ^2+C_2 ^2....+C_n ^2$$ (where $$C_i$$ is the $$i^{th}$$ coefficient of $$(1+x)^n$$ expansion), is:

If $$f(n)=\sum_{s=1}^n \sum_{r=s}^n \:^nC_r \:^rC_s$$, then $$f(3)=$$

The value of $$(n+2)C_02^{n+1}-(n+1)C_12^n+nC_22^{n-1}+....$$ is equal to:
$$(C_r=\:^nC_r)$$

If $$P_n$$ denotes the product of all the coefficients in the expansion of $$(1+x)^n$$, then $$\dfrac {P_{n+1}}{P_n}$$ is equal to:

If $$^{n-1}C_{r}=(k^2-3)\:^nC_{r+1}$$, then $$k\:\:\epsilon$$

The value of $$\displaystyle \:^{50}C_4+\sum_{r=1}^6 \:^{56-r}C_3$$ is

The coefficient of $$x^{53}$$ in the following expansions.
$$\displaystyle \sum_{m=0}^{100} \,^{100}C_m(x-3)^{100-m}\cdot 2^m$$ is