Combinatorics: the maths of counting possibilities

Yesterday, 22 August, at the University of Queensland library, I came across a (fairly old) book which I didn’t know about before, but is very good and very accessible for school students with a will to inquire into the more interesting areas of maths beyond the syllabus.

It’s Ivan Niven’s “The mathematics of choice – how to count without counting”, which is about the maths of counting and tabulating possibilities, usually called “combinatorics”

Combinatorics involves relatively little in the way of formulas, theorems, special notations, complicated chains of calculation, and such. It is mostly about developing systematic and smart ways of mathematical thinking, which are often applicable way beyond combinatorics itself.

Click here for a pdf of the book.

An example of a combinatorics problem (from Niven, p.3):

You work in a building located seven blocks east and eight blocks north of your home. (This is a US city, constructed on a grid pattern, not London!)

How many different routes can you take from home to work, walking only 15 blocks?

(The answer is 6435).