# Pairs of classmates

i have found following problem in my combinatorics book.

The superintendent of the Hardluck Elementary School District suggests that the Board of Education meet a $5$ million(dollar) budget deficit by raising average class sizes, from $30$ to $36$ students, a $20$% increase. A district teacher objects, pointing out that if the proposal is adopted, the potential for a pair of classmates to get into trouble will increase by $45$%. What is the teacher talking about?

i am trying to imagine what is proposal of problem,first line states that if number of pupil increases by 6,then there would be $5$ million dollar deficit ,also number of pair classmates which get in trouble would increases by $45$%,so if for $30$ children,pair of classmates in trouble is about x quantity,then for $36$,it would be $x+0.45*x$,but how is this related to combinatorics?please help me

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I agree the language is confusing. Essentially, the question is asking "Why does $A$ happen 45% more often if we do $B$?". In this case, $A$ is a pair of students gets in trouble, and $B$ is class sizes go from 30 to 36. The combinatorics comes in because 45% seems like a rather high number. If we don't think combinatorially we might (mistakenly) expect a 20% increase, or 40% at the most. – Eric Stucky Jun 30 '12 at 9:30
but does there exist solution generally of this problem? – dato datuashvili Jun 30 '12 at 9:35

That's where combinatorics comes in. You are choosing 2 students from 30. There are 30 ways to choose the first student, and for each of these 30 ways, there are 29 ways to choose the second student, so, all told, $30\times29=870$ ways to get the job done. But we've counted choosing Gauss then Euler as different from choosing Euler then Gauss, and since we're only interested in who the two students are and not in the order in which we chose them, we have to divide by 2. $870/2=435$. – Gerry Myerson Jun 30 '12 at 23:19