maths.free › Combinatorics & Graph Theory › Induction › Introduction

Introduction

A professor decides to liven up the next combinatorics class by giving a door prize. As students enter class (on time, because to be late is a bit insensitive to the rest of the class), they draw a ticket from a box.

Introduction

A professor decides to liven up the next combinatorics class by giving a door prize. As students enter class (on time, because to be late is a bit insensitive to the rest of the class), they draw a ticket from a box. On each ticket, a positive integer has been printed. No information about the range of ticket numbers is given, although they are guaranteed to be distinct. The box of tickets was shaken robustly before the drawing, so the contents are thoroughly mixed, and the selection is done without looking inside the box.

After each student has selected a ticket, the professor announces that a cash prize of one dollar (this is a university, you know) will be awarded to the student holding the lowest numbered ticketfrom among those drawn.

Must the prize be awarded? In other words, given a set of positive integers, in this case the set of ticket numbers chosen by the students, must there be a least one? More generally, is it true that in any set of positive integers, there is always a least one? What happens if there is an enrollment surge and there are infinitely many students in the class and each has a ticket?

Now you No calculator settles this one, but the pieces of it are computable. Try one below, or type your own.

Keep your own working

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Sign Up Login

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

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.

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

More in 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