Dumbledore and mission

There are multiple student groups in the Dumbledore's Army in Hogwarts. Dumbledore wants to send two student groups into the Forbidden Forest to retrieve a powerful weapon. But, in order to break the protection spell on the weapon and retrieve it, at least one person from 'k' number of houses (i.e Gryffindor, Hufflepuff, Ravenclaw, Slytherin) has to cast a spell on it simultaneously. Therefore, at