# Human Nature, Biology, and Social Structure: A Critical Look at WhatScience Can Tell Us About Society

## An Introduction to Mathematical Probability

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After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

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Permutations
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*permutation*is an arrangement of a group of objects in a particular order. The four numbers 5, 6, 7, and 8 can be arranged in twenty-four different ways:

5678 | 5687 | 5768 | 5786 | 5867 | 5876 |

6785 | 6758 | 6587 | 6578 | 6857 | 6875 |

7568 | 7586 | 7856 | 7865 | 7658 | 7685 |

8567 | 8576 | 8657 | 8675 | 8756 | 8765 |

The product of the number 4 x 3 x 2 x 1 = 4! = 24

of choices for each space | permutations. |

symbol n! is read
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n
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factorial
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. By definition, 0!= 1.

If n > 0, n! = 1 x 2 x 3 x . . . x (n-1) x n

nPn = n! | 4P4 = 4! = 1 x 2 x 3 x 4 = 24 |

The number of permutations of six objects taken three at a time is 6 x 5 x 4 = 120.

6P3 = 6 x 5 x 4 = 1 x 2 x 3 x 4 x 5 x 6 = ____6!____ = 120

3 x 2 x 1 | (6-3)! |

____ (n-r)!

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Example 1
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: How many different ways can five books be arranged on a shelf?

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Solution
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: 5P5 = 5! = 1 x 2 x 3 x 4 x 5 = 120

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Example 2
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: How many two digit numbers can be made from the six digits 7, 2, 4, 5, 9, 3 if no digit can be used more than once?

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Solution
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: 6P2 = __ 6__! = 1 x 2 x 3 x 4 x 5 x 6 = 30

(6-2)! | 1 x 2 x 3 x 4 |

*Example 3*: How many integers of three places can be formed from the digits 5, 1, 8, 4 if repetition is allowed?

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Solution
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: 4 x 4 x 4 = 64

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Example 4
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: How many even integers of four places can be formed from the digits 1, 2, 3, 4, 5?

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Solution
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: 4 x 3 x 2 x 2 = 48

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Example 5
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: How many five digit telephone numbers can be made from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9?

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Solution
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: 9 x 9 x 9 x 9 x 9 = 59,049

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Exercises
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:

- ____ 1.) In how many ways can the offices of president, secretary and treasurer be filled from a group of nine people?
- 2.) In how many ways can five girls be arranged in a straight line?
- 3.) In how many ways can seven boys be arranged in a straight line if one particular boy is to be at the beginning of the line, one particular boy is to be in the middle of the line, and one particular boy is to be at the end of the line?
- 4.) How many integers between 10 and 100 can be formed by the digits 1, 2, 3, 4, 5 if no repetition is allowed? How many can be formed if repetition is allowed? 5.) How many odd-numbered integers can be formed by the digits 2, 3, 6, 5, 9, 8 if each digit may be used only once?
- 6.) In how many different ways can the letters of the word number be arranged if each arrangement begins with a vowel?
- 7.) A theater has five entrances. In how many ways can you enter and leave by a different entrance?
- 8.) In how many ways can you mail three letters in six letter boxes if no two are mailed in the same box?
- 9.) Milltown has eight grocery stores and six meat markets. In how many ways can you buy a pound of hot dogs and a bag of flour?
- 10.) Four people enter a bus in which there are six empty seats. In how many ways can the people be seated?