Pick which five boys are participating in the dance.
From those boys selected, one will be the youngest. Pick which girl dances with the youngest selected boy.
Pick which girl dances with the second youngest selected boy.
Continue in this fashion until all selected boys have a designated dancing partner.
This yields a total of $\binom{6}{5}\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3$