Probability and statistics
Worksheet 1 — Sets and counting
1. (a) Write down any set A of size 5. (b) What is the formal notation for all sequences of three elements from A? (c) How many such sequences are there, exactly?
2. How many binary sequences of length 500 are there? 3. A and B are sets with |A|= 3 and |B|= 4. (a) What is the largest size A[B could possibly have? (b) What is the smallest size A[B could possibly have? (c) Repeat for A\B. 4. A donkey, an ox, a goat, and a tiger need to cross a river. They have a boat that can only hold one animal, so they need to go one at a time. How many di↵erent orderings are there?
5. How many sequences of 5 English characters are there?
6. You have 10 good friends, and you want to choose 3 of them to accompany you on a trip. How many groups of three friends can you choose?
7. You have 10 di↵erent beer bottles, and you want to line 5 of them up on your mantelpiece. How many di↵erent arrangements can you make?