13. Romeo, Giulietta, Elena, Paride, Achille, and Ulisse sit on a bench. Giulietta wants to sit next to Romeo and Elena next to Paride. In how many ways can the six arrange themselves from right to left to satisfy them?
(A) 720
(B) 120
(C) 24
(D) 96
(E) 180
(13) The correct answer is $(D)$.

Scanning the bench from right to left, we count how many possible arrangements there are for the following 4 blocks: the couple Romeo and Juliet (RG), the couple Helen and Paris (EP), Achilles (A), and Ulysses (U), initially without considering the order in which the people appear within the two couples.

The 4 blocks can be arranged in $4 \cdot 3 \cdot 2=24$ ways: indeed, there are 4 possible choices for the block in the first position, 3 possible choices for the block in the second position, and 2 possible choices for the block in the third position (the one in the last position is then forced).

For each arrangement of the 4 blocks mentioned above, there are 2 possible orders for the couple (RG) and 2 possible orders for the couple (EP), which can be combined in $2 \cdot 2=4$ possible ways. In conclusion, the total number of arrangements that meet the given conditions is equal to $24 \cdot 4=96$.

Question proposed by Paolo Negrini.