Permutations and Combinations
description
Transcript of Permutations and Combinations
![Page 1: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/1.jpg)
Permutations and Combinations
In this lesson we single out two important special cases of the Fundamental Counting Principle—permutations and combinations.
Goal: Identity when to use permutations and combinations.
![Page 2: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/2.jpg)
Permutation A permutation is one of the different
arrangements of a group of items where order matters.
A permutation of a set of distinct objects is an ordering of these objects.
Anytime you see “order”, plug your numbers into the permutation equation.
Permutations give really big numbers!!
![Page 3: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/3.jpg)
Permutation Example
Some permutations of the letters ABCDWXYZ are
XAYBZWCD ZAYBCDWX DBWAZXYC
YDXAWCZB How many such permutations are
possible? 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
![Page 4: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/4.jpg)
Now, there is a short cut to writing out 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
Its called a factorial, and it looks like an exclamation mark (!).
The number of permutations of n objects is n!.
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
4! = 4 x 3 x 2 x 1 = 24
![Page 5: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/5.jpg)
How many permutations consisting of five letters can be made from these same eight letters? (ABCDWXYZ)
Some are: XYZWC AZDWX AZXYB WDXZB
By the Fundamental Counting Principle, the number of such permutations is
8 x 7 x 6 x 5 x 4 = 6720But there is another shortcut….
![Page 6: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/6.jpg)
Permutation Formula
If a set has n elements, then the number of ways of ordering r elements from the set is denoted by P(n, r).
![Page 7: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/7.jpg)
(n = the number of elements you can choose from; r = how many you are actually going to use)
![Page 8: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/8.jpg)
So, from the question of: How many permutations consisting of five letters can be made from these same eight letters? (ABCDWXYZ)
P(8,5)
![Page 9: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/9.jpg)
Lets try this with students…
![Page 10: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/10.jpg)
Example 1:
A club has nine members. In how many ways can a president, vice president, and secretary be chosen from the members of this club?
Does order matter? Yes, then it is a permutation, we can use
the permutation formula. P(9, 3) =
= 504
![Page 11: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/11.jpg)
Example 2: From 20 raffle tickets in a hat, four tickets
are to be selected in order. The holder of the first ticket wins a car, the second a motorcycle, the third a bicycle, and the fourth a skateboard. In how many different ways can these prizes be awarded?
Does order matter? Yes, then it is a permutation, we can use
the permutation formula. P(20, 4) =
![Page 12: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/12.jpg)
Raffle tickets cont….
P(20,4) =
116,280
![Page 13: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/13.jpg)
Now lets practice….
![Page 14: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/14.jpg)
Combinations
When finding permutations, we are interested in the number of ways of ordering elements of a set. In many counting problems, however, order is not important…
![Page 15: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/15.jpg)
Combinations
A combination of r elements of a set is any subset of r elements from the set (Order does not matter).
If the set has n elements, then the number of combinations of r elements is denoted by C(n, r).
Combinations give smaller numbers!!
![Page 16: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/16.jpg)
Combination Formula
• The key difference between permutations and combinations is order. If we are interested in ordered arrangements, then we are counting permutations; but if we are concerned with subsets without regard to order, then we are counting combinations.
![Page 17: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/17.jpg)
How to tell if order matters… Which one is a permutation and which is a combination? A coach must choose five starters
from a team of 12 players. How many different ways can the coach choose the starters?
How many different ways can the coach select the 1st star, 2nd star, and 3rd star of the game?
![Page 18: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/18.jpg)
Example 1: A club has nine members. In how
many ways can a committee of three be chosen from the members of this club?
Does order matter? No, then it is a combination. C(9,3)=
= 84
![Page 19: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/19.jpg)
Example 2: From 20 raffle tickets in a hat, four
tickets are to be chosen at random. The holders of the winning tickets are to be awarded free trips to the Bahamas. In how many ways can the four winners be chosen?
C(20,4)=
= 4845
![Page 20: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/20.jpg)
Example 3: There are fourteen juniors and three
seniors in the Service Club. The club is to send four representatives to the State Conference. How many different ways are there to select a group of four students to attend the conference?
C(17,4)= 2380
![Page 21: Permutations and Combinations](https://reader035.fdocuments.in/reader035/viewer/2022081503/56816223550346895dd25167/html5/thumbnails/21.jpg)
Now practice…..