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This Lesson (Permutations) was created by by Alwayscheerful(414)
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By the permutations of the letters abc we mean all of their possible arrangements: abc There are 6 permutations for three things. As the number of things (letters in this case) increases, their permutations grow astronomically; for example, if twelve different things are permuted, then the number of their permutations is 479,001,600. This number was derived theoretically from the FUNDAMENTAL PRINCIPLE OF COUNTING. If something can be chosen, or can happen, or be done, in m different ways, and, after that has happened, something else For example, imagine the letters a, b, c, d put into a hat, and let us draw 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. Therefore, there are ab b means that a was chosen first and b second; ba means 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. That is, the number of permutations of 4 different things taken 4 at a time is 4!. 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: EXAMPLE 2 There are 6! permutations of the 6 letters of the word square. a) In how many of them is r the second letter? _ r _ _ _ _ b) In how many of them are q and e next to each other? Answer:a) Let r be the second letter. Then there are 5 ways to fill the first spot, 4 ways to fill the third, 3 to fill the fourth, and so on. There are 5! such permutations. b) Let q and e be next to each other as qe. Then we will be permuting the 5 units qe, s, u a, r. They have 5! permutations. But q and e could be together as eq. Therefore, the total number of ways they can be next to each other is 2· 5! = 240. We have seen that the number of ways of choosing 2 letters from 4 is 4· 3 = 12. This is called The number of permutations of 4 We write this as The lower index 2 indicates the number of factors. The upper index 4 indicates the first factor. For example, = For, there are 8 ways to choose the first, 7 ways to choose the second, and 6 ways to choose the third. In general,
Factorial representation Now, In general, the number of arrangements -- permutations -- of n things take k at a time, can be represented as follows: The upper factorial is the upper index of P, while the lower factorial is the difference of the indices. EXAMPLE 3 Express Answer:10!/6! The upper factorial is the upper index, and the lower factorial is the difference of the indices. When the 6!'s cancel, the numerator becomes This is the number of permutations of 10 different things taken 4 at a time. EXAMPLE 4 Calculate Answer: Hope this helps! This lesson has been accessed 25518 times. |
