In a group of students, 50 speak English, 50 speak French and 50 speak Spanish. Some students speak more than one language. Prove it is possible to divide the students into 5 groups (not necessarily equal), so that in each group 10 speak English, 10 speak French and 10 speak Spanish.

First separate the students (as much as possible) into groups with exactly one student who speaks a particular language.

Then there must be an equal number of students who speak each pair of languages. (For example: AB AB AC AC BC BC) -- simply combine one of each type, (AB AC BC) and then put these smaller groups together to get large groups of 10 each, adding the other groups of one each as necessary.

The reason there must be an equal number of students who speak each pair of languages is because nothing else is possible -- it will either violate the assumption that we now still have the same number of students per language or the assumption that we have no combinations of students which form a group with exactly one student who speaks a particular language.

Posted by Gamer on 2011-12-10 18:28:51