Marinada '21

[lang=hr] Prvih $100$ prirodnih brojeva podijeljeni su nasumično na padajući niz $\{ a_i \}$ i rastući niz $\{ b_i \}$, pri čemu svaki ima $50$ članova, a svaka podjela ima jednaku vjerojatnost da bude izabrana. Koliko iznosi očekivana vrijednost od $\sum^{50}_{i=1} \mathopen| a_i - b_i \mathclose|$ ? [/lang] [lang=en] The first $100$ positive integers are randomly split into a strictly decreasing sequence $\{ a_i \}$ and a strictly increasing sequence $\{ b_i \}$, each with $50$ terms, where each such division has an equal probability of occuring. What is the expected value of $\sum^{50}_{i=1} \mathopen| a_i - b_i \mathclose|$? [/lang]