# Interesting facts and problems to motivate high school combinatorics students

I will give some classes in combinatorics to high school students and I would like to know some facts (and proof) I can show to my students to motivate them to study this beautiful subject.

The problem is I don't know how to calculate these facts and I'm looking for interesting real life problems to solve with my students. So my question is do you know some interesting real life problems I could show and proof to my students?

Thanks

• Real life examples are usually boring. You can consult some combinatorics books to get examples that are interesting (for example, Lovász has a book with lots of exercises and problems). It looks like you're considering counting problems in probability, which wouldn't completely fit into "combinatorics." – Pedro Tamaroff Jul 20 '15 at 3:46
• One of the best things to do is the Birthday Paradox. If you have a group of 20 students, it's coin flip chance that at least two people have the same birthday (that was the first day of class in my college Probability course--nobody shared a birthday though, which will happen but given a couple of classes it's unlikely to happen always). – Jared Jul 20 '15 at 3:47
• Another example with conditional probabilities is the lets make a deal example where it seems there is no benefit to switching your choice (after the first, incorrect, door has been revealed) but, probabilistically there actually is a benefit to switching! – Jared Jul 20 '15 at 3:51
• @Jared: Yes! aka 'The Monty Hall Problem'. This one would be a great one to get the conversation started in a high school math class because it's very surprising. – john Jul 20 '15 at 3:53
• Easy to grasp is the case of derangements. You can easily set up experiments and show that the probability is surprisingly constant. – vonbrand Jul 23 '15 at 14:33