Interesting facts and problems to motivate high school combinatorics students

Question

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.

I'm thinking to talk about this: 15 Things More Likely to Happen than Winning Mega Millions.

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

Answer 1

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!

Answer 2

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).