Lauretta J. Fox
The four numbers 1, 6, 6, 3 can be arranged in 4! ways. Because two of the numbers are alike, several of the arrangements are the same and cannot be distinguished from others. Consider the digits in this way: 1, 6, 6*, 3. The following arrangements can be made:
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16*63 1
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6*36
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166*3
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1636*
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136*6
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1366*
|
|
6*631
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6*613
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6*136
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6*163
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6*316
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6*361
|
|
516*3
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6136*
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6316*
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636*1
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66*13
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66*31
|
|
316*6
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3166*
|
36*16
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36*61
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3616*
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366*1
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Although twenty-four permutations have been formed, there are only twelve distinct arrangements. The number of distinct permutations of four objects when two are alike may be denoted by:
4! = 1 x 2 x 3 x 4 = _24_ = 12
2! 1 x 2 2
The number of distinct permutations of n objects of which s are alike, t are alike, etc. is
n! _
s! t! . . .
Example
: How many different permutations can be made using all the letters of the word Connecticut?
Solution:
The word Connecticut contains eleven letters including three C’s, two N’s, and two T’s. The number of different permutations of these letters is
11! = 3! x 4 x 5 X 6 x 7 x 8 x 9 x 10 x 11
= 1,663,200
Exercises
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1.) How many different permutations can be made using all the letters of the word dinner?
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2.) How many distinct permutations can be made using all the letters of the word (a) challenge (b) banana (c) staff (d) tuition (e) assassination (f) committee?
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3.) How many different seven digit numbers can be made using all the seven digits 3, 3, 3, 4, 4, 5, 5?
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4.) In how many ways can five nickels, three dimes, four pennies and a quarter be distributed among thirteen people so that each person may receive one coin?
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5.) How many signals can be made by raising four red flags, two green flags, and one white flag on a pole at the same time?
