Binomial Theore...

The $$7th$$ term in the expansion of $$\bigg(\dfrac{1}{2}+a\bigg)^8$$ is :

The sum of the coefficients of even powers of $$x$$ in the expansion of
$$ (1+x+x^2+x^3)^5 $$ is

Find the middle term of the expansion of $$\left ( 3x+\frac{1}{2x} \right )^7$$

If $$S$$ be the sum of the coefficients in the expansion of $$(ax+by+-cz)^{n}$$ where $$a, b, c$$ are lengths of the sides of a triangle, then $$\lim_{n\rightarrow \infty }\dfrac{S}{(S^{1/n}+1)^{n}}$$ is

The value of the expression $$\displaystyle \frac{C_1}{C_0}+2\frac{C_2}{C_1}+3\frac{C_3}{C_2}+\ldots\ldots\ldots +n\frac{C_n}{C_{n-1}}$$ is

Arrange the values of $$n$$ in ascending order
A : If the term independent of $$x$$ in the expansion of $$\left(\displaystyle \sqrt{x}-\frac{n}{x^{2}}\right)^{10}$$ is $$405$$
B : If the fourth term in the expansion of $$\left(\displaystyle \frac{1}{n}+n^{\log_{n}10}\right)^{5}$$ is $$1000$$, ( $$ n< 10 $$)
C : In the  binomial expansion of $$(1+x)^{n}$$ the coefficients of  $$5^{\mathrm{t}\mathrm{h}},\ 6^{\mathrm{t}\mathrm{h}}$$ and $$7^{\mathrm{t}\mathrm{h}}$$ terms are in A.P.

If the fourth term in the expansion of $$\displaystyle \left ( \sqrt{x^{\left ( \frac{1}{log x+1} \right )}}+x^{\frac{1}{12}} \right )^6$$ is equal to 200 and x > 1, then x is equal to

The coefficient of $$x^r[0 \le r \le n-1]$$ in the expression of $$(x + 2)^{n-1} + (x+2)^{n-2} .(x+1) + (x+2)^{n-3}. (x+1)^2+...+(x+1)^{n-1}$$ is

Find the sum of the series
$$3.{ _{  }^{ n }{ C } }_{ 0 }-8.{ _{  }^{ n }{ C } }_{ 1 }+13._{  }^{ n }{ { C }_{ 2 } }-18.{ _{  }^{ n }{ { C }_{ 3 } } }+\ldots+(n+1)\quad terms$$