Lauretta J. Fox
A
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
|
Each of these arrangements is a different permutation. To determine the total number of permutations that can be made from four digits using each one only once we indicate a space for each digit __ __ __ __. Any one of the four digits may be placed in the first space. Any of the three remaining digits may occupy the second space. There are two choices left for the third space, and the last digit is placed in the fourth space. The product of the number of choices for each space is the total number of permutations that can be made.
The product of the number 4 x 3 x 2 x 1 = 4! = 24
of choices for each space
|
permutations.
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The total number of permutations that can be formed from n objects using all of them without repetition is n! The
symbol n! is read
n
factorial
. 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
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The symbol nPr represents the number of permutations that can be formed from n objects taken r at a time where r n.
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
Hence nPr = __n!__
____
(n-r)!
Example 1
: How many different ways can five books be arranged on a shelf?
Solution
: 5P5 = 5! = 1 x 2 x 3 x 4 x 5 = 120
Example 2
: 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?
Solution
: 6P2 = __ 6__! = 1 x 2 x 3 x 4 x 5 x 6 = 30
Example 3
: How many integers of three places can be formed from the digits 5, 1, 8, 4 if repetition is allowed?
Solution
: 4 x 4 x 4 = 64
Example 4
: How many even integers of four places can be formed from the digits 1, 2, 3, 4, 5?
Solution
: 4 x 3 x 2 x 2 = 48
Example 5
: How many five digit telephone numbers can be made from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9?
Solution
: 9 x 9 x 9 x 9 x 9 = 59,049
Exercises
:
-
____
1.) In how many ways can the offices of president, secretary and treasurer be filled from a group of nine people?
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2.) In how many ways can five girls be arranged in a straight line?
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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?
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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?
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6.) In how many different ways can the letters of the word number be arranged if each arrangement begins with a vowel?
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7.) A theater has five entrances. In how many ways can you enter and leave by a different entrance?
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8.) In how many ways can you mail three letters in six letter boxes if no two are mailed in the same box?
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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?
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10.) Four people enter a bus in which there are six empty seats. In how many ways can the people be seated?