Let $$X = {({}^{10}{C_1})^2} + 2{({}^{10}{C_2})^2} + 3{({}^{10}{C_3})^2} + ... + 10{({}^{10}{C_{10}})^2}$$,

where $${}^{10}{C_r}$$, r $$ \in $${1, 2, ..., 10} denote binomial coefficients. Then, the value of $${1 \over {1430}}X$$ is ..........

Let $$m$$ be the smallest positive integer such that the coefficient of $${x^2}$$ in the expansion of $${\left( {1 + x} \right)^2} + {\left( {1 + x} \right)^3} + ........ + {\left( {1 + x} \right)^{49}} + {\left( {1 + mx} \right)^{50}}\,\,$$ is $$\left( {3n + 1} \right)\,{}^{51}{C_3}$$ for some positive integer $$n$$. Then the value of $$n$$ is

The coefficient of three consecutive terms of $${\left( {1 + x} \right)^{n + 5}}$$ are in the ratio $$5:10:14.$$ Then $$n$$ =