Counting and Combinations

"Counting" may seem like a simple topic, but there's more to it than you might think.

Here's what you'll find in this section:

Countable -- What does it mean for a set to be countable?  How can I show that there are just as many ordered pairs (pairs such as (1,3) and (-4,11)) as there are integers?  Read this page to find out.

Objects In Boxes -- Several different questions are considered:

Permutations of sets with some indistinguishable objects 

Different orders of MISSISSIPPI, for example; or orders of FACETIOUS keeping vowels in order.

Indistinguishable Objects to Distinguishable Boxes

E.g. Ways to put 5 balls into 3 boxes: C(5+3-1,3-1) = 21

Indistinguishable Objects to Indistinguishable Boxes

Partitions: E.g. 7 can be partitioned into exactly 3 non-empty boxes in 4 ways.

Distinguishable Objects to Indistinguishable Boxes

Ways to distribute n labeled balls into exactly k indistinguishable non-empty boxes

Rectangles within rectangles

Related Pages in this Website

The Peano Postulates -- Proving the properties of natural numbers using the Peano Postulates, which have been formulated so that zero is not included in the set of natural numbers.  (There's quite a debate about this point.)

Is Zero a Natural Number? -- a discussion of the fact that some authors include zero, and others do not.

Introduction to Counting -- explains what mathematicians mean by "counting" -- that is, putting sets in one-to-one correspondence.

Construction -- Construction of sets of numbers, starting with the original Peano Axioms, formulated so that zero is included in the set of natural numbers.

Set Theory -- an introduction to sets, including examples of some standard sets.

Counting Ordered Pairs of Integers -- An explanation of the "square spiral" that puts the set of natural numbers in one-to-one correspondence with the set of rational numbers.

Counting the number of times Friday the 13th occurs.

Counting the number of partitions of n into powers of two such that no power is used more than twice.

The webmaster and author of the Math Help site is Graeme McRae.
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