Binomial Theore...

The coefficient of $$x^5$$ in the expansion of $$(1+x)^{21}+(1+x)^{22}+........+(1+x)^{30}$$ is:

The co efficient of $${ x }^{ 99 }$$ in the polynomial $$\left( x-1 \right) \left( x-2 \right) \left( x-3 \right) .....\left( x-100 \right) $$ is

If the sum of the co-efficient in the expansion of $$(a+b)^n$$ is $$1024$$, then the greatest co-efficient in the expansion is 

If $$n\ge 2$$ then $$3.{ C }_{ 1 }-4.{ C }_{ 2 }+5.{ C }_{ 3 }-......+{ \left( -1 \right)  }^{ n-1 }\left( n+2 \right) .{ C }_{ n }$$ is equal to

For $$2\leq r \leq n, \left( ^{n+1} _r\right)+\left( ^n _{r-1}\right) + \left( ^n _{r-2}\right)$$ is equal to-

The $$6^{th}$$ coefficient in the expansion of $$\left (2x^2 - \dfrac {1}{3x^2}\right)^{10}$$

The sum $$^{10}C_3 + ^{11}C_3 + ^{12}C_3 + .... + ^{20}C_3$$ is equal to

Prove that $$C_0+C_1+C_2+.....C_n=2^n$$

$$\sum\limits_{k = 1}^{n - r} {{}^{n - k}\mathop C\nolimits_r  = {}^x\mathop C\nolimits_y } $$

Find the $$13^{th}$$ terms in the expansion of $$\left(9x-\dfrac{1}{3\sqrt{x}}\right)^{18}, x \neq 0$$.