Binomial Theore...

Number of terms free from radical sign in the expansion of $$\displaystyle \left ( 1+3^{1/3}+7^{1/2} \right )^{10}$$ is

If $${C}_{r}$$ stands for $$^{n}{C}_{r}$$, then the sum of the series $$\displaystyle\frac { 2\left( \frac { n }{ 2 }  \right) !\left( \frac { n }{ 2 }  \right) ! }{ n! } \left[ { C }_{ 0 }^{ 2 }-2{ C }_{ 1 }^{ 2 }+3{ C }_{ 2 }^{ 2 }-...+{ \left( -1 \right)  }^{ n }\left( n+1 \right) { C }_{ n }^{ 2 } \right] $$ where $$n$$ is an even positive integer, is

$$(r + 1)^{th}$$ term in the expansion of $$(x + a)^n$$ will be

If coefficients of $$x^n$$ in $$(1+x)^{101}(1-x+x^2)^{100}$$ is non-zero then $$n$$ cannot be of the form

$$\displaystyle \binom{n}{o}+\binom{n}{1}+\binom{n}{2}+.........+\binom{n}{n}=$$

In the expansion of $$(1 + x)^n$$, the sum of coefficients of odd powers of $$x$$ is

If $$(1+x)^n=C_0+C_1x+C_2x^2+.....+C_nx^2$$, then the value of $$C_0+C_2+C_4+ .....$$ is

$$^{10}C_1+^{10}C_3+^{10}C_5+^{10}C_7+^{10}C_9=$$

The value of $$3{\;}^nC_0-8{\;}^nC_1+13 {\;}^nC_2-18 {\;}^nC_3+.....+n$$ terms is

Find the middle terms in the expansion of $$\displaystyle \left ( 2x^{2}-\frac{1}{x} \right )^{7}$$