probability day 3 - permutations and combinations
DESCRIPTION
TRANSCRIPT
![Page 1: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/1.jpg)
Factorial Notation The expression 6 × 5 × 4 × 3 × 2 × 1 = can be written as 6!, which is read as “six factorial.”
In general, n! is the product of all the counting numbers beginning with n and counting backwards to 1.
We define 0! to be 1. Factorial on your TI calculator.
![Page 2: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/2.jpg)
Example :Find the value of each expression:a) 3!
b) 0!
c) 3! + 2!
d)
![Page 3: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/3.jpg)
Fundamental Counting Principle:If one activity can occur in any of m ways and, following this, a second activity can occur in any of n ways, then both activities can occur in the order given in m*n ways.
![Page 4: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/4.jpg)
Permutation Formula an arrangement of objects in some specific orderIn general P(n, r) means the number of permutations of n items arranged r at a time.
The formula for permutation is
Permutations on your TI calculator
Note:nPn = n!
3P3 = 3! = 3*2*1 = 6
![Page 5: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/5.jpg)
Words used in permutation problems:• arrangement• line up• president, vice president, secretary• 1st, 2nd, 3rd place
Example :A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different arrangements of these letters can be made if no letter is used more than once?
![Page 6: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/6.jpg)
Permutations with repetitionIf we want to arrange items when there are more than one of the same item, we need to divide by the number of identical items:Example:Find the number of arrangements of the letters that can be formed from the letters IDENTITY, using each letter
Solution:Example: Find the number of arrangements of letters that can be formed from the letters:
MINIMUMSolution:
![Page 7: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/7.jpg)
CombinationsAn arrangement of objects in which the order is not important is called a combination. This is different from permutation where the order matters. For example, suppose we are arranging the letters A, B and C. In a permutation, the arrangement ABC and ACB are different. But, in a combination, the arrangements ABC and ACB are the same because the order is not important.
![Page 8: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/8.jpg)
The number of combinations of n things taken r at a time is written as C(n, r).
The formula is given by:
Combinations on your TI calculator
![Page 9: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/9.jpg)
Example: In how many ways can a coach choose three swimmers from among five swimmers?
Words used in combination problems:• committee• group• team
![Page 10: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/10.jpg)
![Page 11: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/11.jpg)
Example 6:There are 5 red and 4 white marbles in an urn. A marble is drawn from the urn and not replaced. Then, a second marble is drawn. a. In how many ways can a red marble and a white marble be drawn in that order?
b. In how many ways can a red marble and a white marble be drawn in either order?
Example 7:An urn contains three white balls and four red balls. Two balls are chosen at random. How many ways can you chose at least one of the red balls?
![Page 12: Probability Day 3 - Permutations and Combinations](https://reader034.vdocuments.net/reader034/viewer/2022051107/540acf7d8d7f72ab0c8b46b3/html5/thumbnails/12.jpg)