# 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

To calculate those facts, you would need some statistics. For example, to calculate the death by vending machine problem, you would need statistics on how many people each year use vending machines, and how many die from vending machines. For high school students, I think it would be cool to arrange the seats in a circle and have them experiment on the number of of different ways they could sit in it, and then show them the possibilities using combinatorics. Another proof you could do would be of Pascal's Rule. It is a simple proof requiring little knowledge of combinatorics, and would be easy for the high school students to understand. Good Luck!

Count the number of ways of placing them in line, in a circle; how many ways to form a line of, say, 5 of the students. How many ways to choose, say, 5 of them gives binomial coefficients. How many ways are there to make up a 5-person group out of 2 boys and 3 girls, this generalizes to Vandermonde's identity. Fool around with a 3-way split of the class, and get the respective identity.

More challenging is the number of ways of splitting the class up into say 4 groups (Stirling numbers of the second kind). Or consider the number of possible results in a competition, where ties are possible (i.e., several tie for first, second, ... places; this is ordered Bell numbers).