Stats 110: Practice 1

Exercises from Chapter 1 of Blitzstein and Hwang (2019)

  1. There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces?

Let the players be numbered \(1..20\). Each permutation of the list can be considered a pairing, if you draw bars like so:

\[ 1 \, 3 \, | \, 4 \, 6 \, | \, 2 \, 5 \,|\, ... \] The total number of permutations is \(20!\)

Each pair like \(|1\,3|\) is two people sitting down at a table for a game, with white and black pieces respectively. The order within a pair matters. However, the order of tables themselves doesn’t matter. There are \(10!\) ways to permute the tables, and all those must be considered equivalent.

Thus the total number of ways to match 20 people up is: \(\frac{20!}{10!}\).

  1. You are ordering two pizzas. A pizza can be small, medium, large, or extra large, with any combination of 8 possible toppings (getting no toppings is allowed, as is getting all 8). How many possibilities are there for your two pizzas?

References

Blitzstein, Joseph K., and Jessica Hwang. 2019. Introduction to Probability. Second edition. Boca Raton: CRC Press.