Binomial Theore...

If the third term in the expansion of $$\displaystyle (\frac{1}{x}+x^{\log_{10}x})^{5} $$ is $$1,000$$, then $$x$$-equals

In the expansion of $$\displaystyle (1+x)^{23}$$, if $$\displaystyle r^{th}$$ ,$$\displaystyle (r+1)^{th}$$ , and $$\displaystyle (r+2)^{th}$$ terms are in A.P., then value of  $$\displaystyle ^{23}C_{r}$$ equals

The number of integral terms in the expansion of $$\displaystyle \left ( \sqrt{3}+\sqrt[5]{5} \right )^{256}$$ is

If $$\displaystyle C_{r}=^{n}C_{r}$$ and $$\displaystyle (C_{0}+C_{1})(C_{1}+C_{2})...(C_{n-1}+C_{n})=k$$ $$\displaystyle (C_{0} C_{1}C_{2}...C_{n})$$ then the value of $$ \displaystyle k$$ equals

If $$\displaystyle P$$ be the sum of odd term and  $$\displaystyle Q$$ that of even terms in the expansion of  $$\displaystyle (x+a)^{n}$$ , then the value of  $$\displaystyle [(x+a)^{2n}-(x-a)^{2n}]$$ equals

$$^{ n+1 }{ { C }_{ 2 }^{  } }+2\left[ _{  }^{ 2 }{ { C }_{ 2 }^{  } }+^{ 3 }{ { C }_{ 2 }^{  } }+^{ 4 }{ { C }_{ 2 }^{  } }+...+^{ n }{ { C }_{ 2 }^{  } } \right] =$$

Number of terms in the expansion of $$ \left ( 1-x \right )^{51}\left ( 1+x+x^{2} \right )^{50}$$ of

If $${C_{r}}^{13}$$ denoted by $$C_{r}$$ then value of $$c_{1}+c_{5}+c_{7}+c_{9}+c_{11}$$ is equal to

If coefficient of $$x^{100}$$ in $$1+\left ( 1+x \right )\left ( 1+x \right )^{2}+.....+\left ( 1+x \right )^{n}\left ( if\:n \geq 100\right )$$ is $$C_{101}^{201}$$ then the value of n equals

If $$C_{0},C_{1},C_{2},.....C_{n},$$ are binomial coefficients,then $$\displaystyle\sum_{k= 0}^{n} C_{k}\:\sin kx \cos \left ( n-k \right )x$$ equals