# MATH 1P98 Chapter Notes - Chapter 4: Natural Number, Graphing Calculator, Binomial Theorem

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2 Feb 2016

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Stats: Counting Techniques

Definitions

Factorial

A positive integer factorial is the product of each natural number up to and including the

integer.

Permutation

An arrangement of objects in a specific order.

Combination

A selection of objects without regard to order.

Tree Diagram

A graphical device used to list all possibilities of a sequence of events in a systematic

way.

Stats: Counting Techniques

Fundamental Theorems

Arithmetic

Every integer greater than one is either prime or can be expressed as an unique product of prime

numbers

Algebra

Every polynomial function on one variable of degree n > 0 has at least one real or complex zero.

Linear Programming

If there is a solution to a linear programming problem, then it will occur at a corner point or on a

boundary between two or more corner points

Fundamental Counting Principle

In a sequence of events, the total possible number of ways all events can performed is the

product of the possible number of ways each individual event can be performed.

The Bluman text calls this multiplication principle 2.

Factorials

If n is a positive integer, then

n! = n (n-1) (n-2) ... (3)(2)(1)

n! = n (n-1)!

A special case is 0!

0! = 1

Permutations

A permutation is an arrangement of objects without repetition where order is important.

Permutations using all the objects

A permutation of n objects, arranged into one group of size n, without repetition, and order being

important is:

nPn = P(n,n) = n!

Example: Find all permutations of the letters "ABC"

ABC ACB BAC BCA CAB CBA

Permutations of some of the objects

A permutation of n objects, arranged in groups of size r, without repetition, and order being

important is:

nPr = P(n,r) = n! / (n-r)!

Example: Find all two-letter permutations of the letters "ABC"

AB AC BA BC CA CB

Shortcut formula for finding a permutation

Assuming that you start a n and count down to 1 in your factorials ...

P(n,r) = first r factors of n factorial