COUNTING PRINCIPLES
A group of numbers arranged in a specific order is called a sequence. A sequence can be either finite or infinite, depending upon whether or not it has a last term.
Example 1:
An example of a finite sequence is
8, 12, 16, 20, 24.
An example of an infinite sequence
is 11.1, 11.2, 11.3, 11.4, . . .
The nth term of a series is found by adding the first n terms of the associated sequence. An arithmetic series is a series in which the next term is found by addition. A geometric series is a series formed by multiplication. The symbol can be used to indicate a series.
Example 2:
An example of an arithmetic series is
3 + 6 + 9 + 12 + 15 + 18 + 21 + 24.
An example of a geometric series is
2 + 10 + 50 + 250 + . . . + 2(5n), or
In general, an arithmetic series can be represented and a geometric series can be represented as where a is the first term, d is the common difference, and r is the common ratio of two successive terms.
Permutations
When considering the number of possible arrangements of a set of elements in a specific order, we have a permutation.
To deal with permutations, an understanding of the factorial is necessary. A factorial is denoted by a whole number followed by the "exclamation" symbol !. When this symbol is used, the product of natural numbers from 1 to a given whole number is found. Zero factorial, 0!, is defined to be 1. That is, 0! = 1.
Example 3:
10! = 10 9 8 7 6 5 4 3 2 1
10! = 3,628,800
n! = n(n - 1)(n - 2)(n - 3) . . . 1
The number of permutations in a set of elements of n things taken r at a time is indicated as . The formula for finding n things taken r at a time is .
Example 4: In how many different ways can an officer be elected if a group of | 10 people are choosing a president, a vice-president, and a secretary? | All members of the group are willing to take any office. | |
Permutations Formulas n things taken r at a time:
n things taken n at a time:
Circular permutation of n things taken r at a time:
Circular permutations of n things taken n at a time:
Combinations
Combinations occur when a set of elements is a subset of a given set. In combinations, order does not matter.
Combination Formula:
Example 5: George has decided to read 4 books of the 27 books on the
