the permutation of 9
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There are 6 permutations of three different things. As the number of things (letters) increases, their permutations grow astronomically. For example, if twelve different things are permuted, then the number of their permutations is 479,001,600.
Now, this enormous number was not found by counting them. It is derived theoretically from the Fundamental Principle of Counting:
For example, imagine putting the letters a, b, c, d into a hat, and then drawing two of them in succession. We can draw the first in 4 different ways: either a or b or c or d. After that has happened, there are 3 ways to choose the second. That is, to each of those 4 ways there correspond 3. Therefore, there are 4· 3 or 12 possible ways to choose two letters from four.
abbacada ac bc cb db ad bd cd dc
ab means that a was chosen first and b second; bameans that b was chosen first and a second; and so on.
Let us now consider the total number of permutations of all four letters. There are 4 ways to choose the first. 3 ways remain to choose the second, 2 ways to choose the third, and 1 way to choose the last. Therefore the number of permutations of 4 different things is
4· 3· 2· 1 = 24
Thus the number of permutations of 4 different things taken 4 at a time is 4!. (See Topic 19.)
(To say "taken 4 at a time" is a convention. We mean, "4! is the number of permutations of 4 different things taken from a total of 4 different things.")
In general,
The number of permutations of n different things taken n at a time
is n!.
Example 1. Five different books are on a shelf. In how many different ways could you arrange them?
Answer. 5! = 1· 2· 3· 4· 5 = 120
Now, this enormous number was not found by counting them. It is derived theoretically from the Fundamental Principle of Counting:
For example, imagine putting the letters a, b, c, d into a hat, and then drawing two of them in succession. We can draw the first in 4 different ways: either a or b or c or d. After that has happened, there are 3 ways to choose the second. That is, to each of those 4 ways there correspond 3. Therefore, there are 4· 3 or 12 possible ways to choose two letters from four.
abbacada ac bc cb db ad bd cd dc
ab means that a was chosen first and b second; bameans that b was chosen first and a second; and so on.
Let us now consider the total number of permutations of all four letters. There are 4 ways to choose the first. 3 ways remain to choose the second, 2 ways to choose the third, and 1 way to choose the last. Therefore the number of permutations of 4 different things is
4· 3· 2· 1 = 24
Thus the number of permutations of 4 different things taken 4 at a time is 4!. (See Topic 19.)
(To say "taken 4 at a time" is a convention. We mean, "4! is the number of permutations of 4 different things taken from a total of 4 different things.")
In general,
The number of permutations of n different things taken n at a time
is n!.
Example 1. Five different books are on a shelf. In how many different ways could you arrange them?
Answer. 5! = 1· 2· 3· 4· 5 = 120
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