24 ## PERMUTATIONS AND COMBINATIONSThe Fundamental Principle of Counting Factorial representation of permutations Factorial representation of combinations BY THE PERMUTATIONS of the letters
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: If something can be chosen, or can happen, or be done, in For example, imagine putting the letters
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 Thus the number of permutations of 4 different things taken 4 at a time is 4!. (Topic 19.) (To say "taken 4 at a time" is a convention. We mean, "4! is the number of permutations of all 4 of 4 different things.") In general, The number of permutations of n different things taken n at a time Example 1. Five different books are on a shelf. In how many different ways could you arrange them?
Example 2. There are 6! permutations of the 6 letters of the word a) In how many of them is b) In how many of them are
a) Let b) Let Permutations of less than all We have seen that the number of ways of choosing 2 letters
from 4 is 4
We will symbolize 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 We saw in the Topic on factorials,
5! is a factor of 8!, and therefore the 5!'s cancel. Now, 8
In general, the number of arrangements—permutations—of
The upper factorial is that of the upper index of
Example 3. Express
The upper factorial is the upper index, and the lower factorial is the difference of the indices. When the 6!'s cancel, the fraction reduces to 10 This is the number of permutations of 10 different things taken 4 at a time.
Example 4. Calculate
Problem 1. Write down all the permutations of To see the answer, pass your mouse over the colored area.
Problem 2. How many permutations are there of the letters
4! = 1 Problem 3. How many different arrangements (permutations) are there of the digits 34567?
5! = 1 Problem 4. a) If the five letters b) After one of them has been drawn our, in how many ways could you
c) Therefore, in how many ways could you draw two letters?
5 This number is denoted by d) What is the meaning of the symbol It is the number of permutations of 5 different things taken 3 at a time. e) Evaluate Problem 5. Evaluate a)
c) Problem 6. Express with factorials.
Problem 7. a) How many different arrangements are there of the letters of the word 7! = 5,040 b) How many of those arrangements have
Set c) How many have The same. 6!. d) How many will have
Begin by permuting the 5 things— Problem 8. a) How many permutations are there of the digits 01234? 5! = 120 b) How many 5-digit numbers can you make of those digits, in which b) the first digit is not 0, and no digit is repeated?
Since 0 cannot be first, remove it. Then there will be 4 ways to choose the first digit. Now replace 0. It will now be one of 4 remaining digits. Therefore, there will be 4 ways to fill the second spot, 3 ways to fill the third, and so on. The total number of 5-digit numbers, then, is 4 c) How many 5-digit
Again, 0 cannot be first, so remove it. Since the number must be odd, it must end in either 1 or 3. Place 1, then, in the last position. _ _ _ _
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