Common themes in counting

Rebecca — Thu, 15 May 2014 12:00:10 +0000

Break into chunks and then add back together.
Break into tasks and then multiply together.
Do via the back door – find exactly when you don’t want and what’s left over is what you do want. Combining this with DeMorgan’s Laws can be powerful.
- Counting: find the total number of ways to accomplish a task and subtract the number of ways that don’t meet your criteria.
- Probability: take 1 and subtract the probability of what you want not happening.
Restrict your viewpoint: if you have 5 pencils in 750 million writing utensils, and you want to know how many ways there are to choose a sample of 3 of the 5 pencils, ignore the rest of the 750 million and compute 5 choose 3.
Get rid of overlap: the inclusion-exclusion principle shows up in many different guises.

Counting and Probability Quiz

Rebecca — Thu, 27 Mar 2014 12:00:43 +0000

A professor is writing a multiple choice question. The question will show three graphs and ask which of them are connected, and the answer choices will be of the form “I and II only”, “none of them”, “all of them”, “III only”, etc. The professor uses a program to generate the answer selections, which guarantees the correct answer will be one of the options but otherwise randomly assigns answer choices to the letters a) through e).
1. How many different versions of the problem could the computer generate?
2. What is the probability the correct answer is c) in the version generated?
Suppose you have a Minesweeper board which is 10×10 squares large and on which you will place 20 mines.
1. To find the number of board layouts, would you use combination or permutation? Find the number of board layouts.
2. Suppose now that you want to consider rotations and reflections of the board as “the same” layout. If counting the layouts under this condition is simple, find the number of board layouts; if it is complicated, explain why and describe the computations that must be made.