The upper factorial is that of the upper index of P, while the lower factorial is the difference of the indices.Įxample 3. We see, then, that 8 P 3 can be expressed in terms of factorials as follows: The upper index 4 indicates the first factor.įor example, 8 P 3 means: "The number of permutations of 8 different things taken 3 at a time." 8 P 3įor, there are 8 ways to choose the first, 7 ways to choose the second, and 6 ways to choose the third.ĥ! is a factor of 8!, and therefore the 5!'s cancel. The lower index 2 indicates the number of factors. "The number of permutations of 4 different things taken 2 at a time." We have seen that the number of ways of choosing 2 letters Therefore, the total number of ways they can be next to each other is 2 Then we will be permuting the 5 things qe, s, u a, r. There are 5! such permutations.ī) Let q and e be next to each other as qe. After that has happened, there are 4 ways to fill the third, 3 to fill the fourth, and so on. ![]() Then there are 5 ways to fill the first spot. There are 6! permutations of the 6 letters of the word square.Ī) In how many of them is r the second letter? _ r _ _ _ _ī) In how many of them are q and e next to each other?Ī) Let r be the second letter. In how many different ways could you arrange them?Įxample 2. The number of permutations of n different things taken n at a timeĮxample 1. We mean, "4! is the number of permutations of all 4 of 4 different things.") (To say "taken 4 at a time" is a convention. Thus the number of permutations of 4 different things taken 4 at a time is 4!. Therefore the number of permutations of 4 different things is 3 ways remain to choose the second, 2 ways to choose the third, and 1 way to choose the last. Let us now consider the total number of permutations of all four letters. abĪb means that a was chosen first and b second ba means that b was chosen first and a second and so on. 3 or 12 possible ways to choose two letters from four.That is, to each of those possible 4 there will correspond 3. After that has happened, there will be 3 ways to choose the second. We can draw the first in 4 different ways: either a or b or c or d. ![]() If something can be chosen, or can happen, or be done, in m different ways, and, after that has happened, something else can be chosen in n different ways, then the number of ways of choosing both of them is m įor example, imagine putting the letters a, b, c, d into a hat, and then drawing two of them in succession.
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