Marinada '25

[lang=hr] Neka se $20$ dečkiju i $10$ cura poreda u red. Neka $S$ bude broj puta kada dečko i cura stoje jedno do drugoga. Kolika je očekivana vrijednost od $S$ kroz sve permutacije? Zapiši rješenje u obliku $\frac{a}{b}$ gdje su $a$ i $b$ relativno prosti cijeli brojevi te kao odgovor na ovo pitanje upiši $a+b$. [/lang] [lang=en] Twenty boys and ten girls are arranged in a row. Let \(S\) be the number of times a boy and a girl stand next to each other (i.e., the number of adjacent pairs consisting of one boy and one girl). What is the expected value of \(S\) over all permutations? Write the solution in the form \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers, and as the final answer to this question enter \(a+b\). [/lang]