Problems

How many ways can the numbers \(1,1,1,1,1,2,3,\dots,9\) be listed in such a way that no \(1\)’s are adjacent? The number 1 appears five times and each of 2 to 10 appear exactly once.

John’s local grocery store sells 7 kinds of vegetable, 7 kinds of meats, 7 kinds of grains and 7 kinds of cheeses. John would like to plan the entire week’s dinners so that exactly one ingredient of each type is used per meal and no ingredients repeat during the week. How many ways can John plan the dinners?

Suppose there is an \(7 \times 7\) grid. We would like to travel from the bottom left corner to the top right corner in exactly 14 steps. A step is from one point on the grid to another point via a segment of length 1. How many paths are there? The picture below shows one possible path on the grid.

In an office, at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile of the secretary’s in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order \(1,2,3,4,5,6,7,8,9\). While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning’s typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Base upon the above information, how many such after-lunch orders are possible? (That there are no letters left to be typed is one of the possibilities.)

In this sheet, we will look at basic counting problems. The fundamental principle is quite simple. If you have to make two independent choices to make, the number of options for making both choices is calculated by multiplying the number of options for each choice.

A problem one immediately runs into is that of overcounting. This means we counted the same thing more than once. In the examples and problems today, you will see various ideas we can use to correct for overcounting or avoiding it.