### Binomial Theorem

goals
In the expansion of $(1 + x)^n$ by the increasing powers of $x$, the third term is four times as great as the fifth term, and the ratio of the fourth term to the sixth is $\dfrac {40}{3}$. Find $n$ and $x$.

If log 1001= 3.000434, find the number of digits in ${1001^{101}}$

Simplify the binomial $\displaystyle\, \left ( \frac{x + 1}{x^{2/3} - x^{1/3} + 1} - \frac{x -1}{x - x^{1/2}} \right )^{10}$ and find the term of its expansion which does not contain $x$.
Prove that the coefficient of ${ x }^{ p }$ in the expansion of ${ \left( { x }^{ 2 }+\dfrac { 1 }{ x } \right) }^{ 2n }$ is $\dfrac { \left( 2n \right) ! }{ \left( \dfrac { 4n-p }{ 3 } \right) !\left( \dfrac { 2n+p }{ 3 } \right) ! }$