Binomial Theore...

In the expansion of $$\left (5^{ \tfrac {1}{2}}+7^{\tfrac {1}{8}}\right )^{1024}$$, the number of integral terms is

Value of
$$S={ _{  }^{ n }{ C } }_{ r }+3({ _{  }^{ n-1 }{ C } }_{ r })+5({ _{  }^{ n-2 }{ C } }_{ r })+...+ $$ upto $$\quad (n-r+1)\quad terms$$

The third term from the end in the expansion of $$\displaystyle\left(\frac{4x}{3y}-\frac{3y}{2x}\right)^9$$ is

Find the value(s) of k such that the term independent of x in $$\displaystyle\left(3x^2+\frac{k}{2x}\right)^6$$ is 135.

Maximum sum of the coefficients in the expansion of $$ (1 - x \sin \theta+ x^{2} )^{n} $$ is

Find the coefficient of $$x^4$$ in the expansion of $$\left(2x^2+\frac{3}{x^3}\right)^7$$

The sum of the series $$\frac{1}{1\times 2}^{25}C_0 + \frac{1}{2\times 3}^{23}C_1+\frac{1}{3\times 4}^{25}C_2+...... + \frac{1}{26\times 27}^{25}C_{25}$$

The number of terms in the expansion of $$(1 + 5\sqrt 2x)^9 + (1-5\sqrt 2x)^9$$ is:

The number of rational terms in the expansion of $$\left(x^{\displaystyle\frac{1}{5}}+y^{\displaystyle\frac{1}{10}}\right)^{45}$$ is