Binomial Theore...

If sum of all the coefficients in the expansion of $$(x^{3/2} + x^{1/3})^n$$ is 128, then the coefficient of $$x^5$$ is

The sum of the coefficients in the expansion of $$(x + 2y + z)^{10}$$ is

In the expansion of $$\displaystyle \left [ 7^{1/3}+11^{1/9} \right ]^{6561}$$, the number of terms free from radicals is

In the expression of $$\displaystyle \left ( 3-\sqrt{\frac{17}{4}+3\sqrt{2}} \right )^{15}$$, the 11th term is a

The total number of distinct terms in the expansion of $$\displaystyle \left ( x+a \right )^{100}+\left ( x-a \right )^{100}$$ after simplification is

If $$p+q=1$$, then the value of $$\displaystyle \sum _{ r=0 }^{ 15 }{ { _{  }^{ 15 }{ C } }_{ r } } { p }^{ 15-r }{ q }^{ r }$$

The value of $$ \sum _{ r=1 }^{ 10 }{ { r }^{ 2 }.\cfrac { { _{  }^{ 26 }{ C } }_{ r } }{ { _{  }^{ 26 }{ C } }_{ r-1 } }  } $$ is-

If $${ \left( { x }^{ 2 }+\cfrac { 1 }{ { x }}  \right)  }^{ n }$$ has exactly one middle term which is equal to $$\alpha.{x}^{3}$$ then the value of $$(\alpha+n)$$ is-         ($$n\in N$$)

If the second term of the expansion $$\displaystyle \left [ a^{1/13}+\frac{a}{\sqrt{a^{-1}}} \right ]^{n}\: \: is\: \: 14a^{5/2}$$, then the value of $$\displaystyle \frac{^{n}{C}_{3}}{^{n}{C}_{2}}$$ is

The value of the expression $$\displaystyle \left ( \sum_{r=0}^{10} \ ^{10}C_{r} \right )\left ( \sum_{k=0}^{10}\left ( -1 \right )^{k}\dfrac{^{10}\textrm{C}_{k}}{2^{k}} \right )$$ is: