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Introduction

We start this chapter with an elementary example. Example Let X be the set of 63 students in an applied combinatorics course at a large technological university.

Introduction

We start this chapter with an elementary example.

Example

Let \(X\) be the set of \(63\) students in an applied combinatorics course at a large technological university. Suppose there are \(47\) computer science majors and \(51\) male students. Also, we know there are \(45\) male students majoring in computer science. How many students in the class are female students not majoring in computer science?

Solution

Although the Venn diagrams that you've probably seen drawn many times over the years aren't always the best illustrations (especially if you try to think with some sort of scale), let's use one to get started. In , we see how the groups in the scenario might overlap.

Now we can see that we're after the number of students in the white rectangle but outside the two shaded ovals, which is the female students not majoring in computer science. To compute this, we can start by subtracting the number of male students (the blue region) from the total number of students in the class and then subtracting the number of computer science majors (the yellow region). However, we've now subtracted the overlapping region (the male computer science majors) twice, so we must add that number back. Thus, the number of female students in the class who are not majoring in computer science is \[\begin{aligned}\end{aligned}\]

From these examples, you should start to see a pattern emerging that leads to a more general setting. In full generality, we will consider a set \(X\) and a family \(\mathcal{P}=\{P_1,P_2,\dots,P_m\}\) of properties. We intend that for every \(x\in X\) and each \(i=1,2,\dots,m\), either \(x\) satisfies \(P_i\) or it does not. There is no ambiguity. Ultimately, we are interested in determining the number of elements of \(X\) which satisfy none of the properties in \(\mathcal{P}\). In , we could have made property \(P_1\) is a computer science major and property \(P_2\) is male. Then the number of students satisfying neither \(P_1\) nor \(P_2\) would be the number of female students majoring in something other than computer science, exactly the number we were asked to determine. What would the properties \(P_1\) and \(P_2\) be for ?

Let's consider three examples of larger sets of properties. These properties will come back up during the remainder of the chapter as we apply inclusion-exclusion to some more involved situations. Recall that throughout this book, we use the notation \([n]\) for the set \(\{1,2,\dots,n\}\) when \(n\) is a positive integer.

Condensed — the full section is in Keller & Trotter, Applied Combinatorics.

Practice (1)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Referring to the properties \(P_i\) as described in , list the integers in \([15]\) that satisfy \(P_{14}\).

Symbols used here

i
imaginary unit
i² = −1.
n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.

Questions people ask

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges) — not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Ipprova tiegħek stess

Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

Aktar fil Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion–exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs