Expected Value 2

For $1\leq k \leq n$ , let $E_{k,n}$ be the expected value of the k-th smallest number of n numbers chosen randomly and uniformly on the unit interval $[0,1]$ . Let

$A_n = \# \Bigg (\bigcup\limits_{j=1}^{n} \bigcup\limits_{i=1}^{j} \{ E_{i,j} \} \Bigg),$

where $\#(S)$ is the size (cardinality) of the finite set $S$ .

For some positive integers $a, b, c$ ,

$\underset{n\to \infty }{\lim} \frac{A_n}{n^a} = \frac{b}{{\pi}^c}.$

Submit $a + b + c$ .