Post on 29-Jan-2018
Section 4-6Permutations of a Set
Essential Question
How do you find the number of permutations of a set?
Where you’ll see this:
Cooking, travel, music, sports, games
Vocabulary
1. Permutation:
2. Factorial:
3. n Pr :
Vocabulary
1. Permutation: An arrangement of items in a particular order
2. Factorial:
3. n Pr :
Vocabulary
1. Permutation: An arrangement of items in a particular orderORDER IS IMPORTANT!!!
2. Factorial:
3. n Pr :
Vocabulary
1. Permutation: An arrangement of items in a particular orderORDER IS IMPORTANT!!!
2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)...(3)(2)(1)
3. n Pr :
Vocabulary
1. Permutation: An arrangement of items in a particular orderORDER IS IMPORTANT!!!
2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)...(3)(2)(1) 5! = (5)(4)(3)(2)(1)
3. n Pr :
Vocabulary
1. Permutation: An arrangement of items in a particular orderORDER IS IMPORTANT!!!
2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)...(3)(2)(1) 5! = (5)(4)(3)(2)(1) = 120
3. n Pr :
Vocabulary
1. Permutation: An arrangement of items in a particular orderORDER IS IMPORTANT!!!
2. Factorial: The number of permutations of n items: n(n - 1)(n - 2)...(3)(2)(1) 5! = (5)(4)(3)(2)(1) = 120
3. n Pr : n permutations taken r at a time; n is the number of different items and r is the number of items taken at a time
Example 1
How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
Example 1
How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
Example 1
How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
3
Example 1
How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
3 2
Example 1
How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
3 2 1
Example 1
How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
3 2 1
(3)(2)(1)
Example 1
How many 3-digit numbers can be formed from the digits 4, 7, and 9 if no repetitions are allowed?
3 2 1
(3)(2)(1) = 6 numbers
Permutation Formula
Permutation Formula
n Pr =
n!(n− r )!
Permutation Formula
n Pr =
n!(n− r )!
n is the number of different items and r is the number of items taken at a time
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2 =
4!(4−2)!
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2 =
4!(4−2)!
=4!2!
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2 =
4!(4−2)!
=4!2!
=(4)(3)(2)(1)
(2)(1)
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2 =
4!(4−2)!
=4!2!
=(4)(3)(2)(1)
(2)(1)
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2 =
4!(4−2)!
=4!2!
=(4)(3)(2)(1)
(2)(1)
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2 =
4!(4−2)!
=4!2!
=(4)(3)(2)(1)
(2)(1) =12
Four letters are taken two at a time. This is the permutation of finding as many two-letter groupings of the four letters.
Example 2
n Pr =
n!(n− r )!
4 P2 =
4!(4−2)!
=4!2!
=(4)(3)(2)(1)
(2)(1) =12
12 two-letter groupings can be made
Example 3How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and
9 if no repetitions are allowed?
Example 3
n Pr =
n!(n− r )!
How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
Example 3
n Pr =
n!(n− r )!
6 P3
How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
Example 3
n Pr =
n!(n− r )!
6 P3 =
6!(6−3)!
How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
Example 3
n Pr =
n!(n− r )!
6 P3 =
6!(6−3)!
=6!3!
How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
Example 3
n Pr =
n!(n− r )!
6 P3 =
6!(6−3)!
=6!3! = (6)(5)(4)
How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
Example 3
n Pr =
n!(n− r )!
6 P3 =
6!(6−3)!
=6!3! = (6)(5)(4) =120
How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
Example 3
n Pr =
n!(n− r )!
6 P3 =
6!(6−3)!
=6!3! = (6)(5)(4) =120
How many 3-digit numbers can be formed from the digits 4, 5, 6, 7, 8, and 9 if no repetitions are allowed?
120 3-digit number can be formed
Understanding Break
1. What is the short way of writing (6)(5)(4)(3)(2)(1)?
2. Is (2!)(3!) the same as 6! ?
3. Which of the following is the value of 0! ?a. 0 b. 1 c. -1 d. ±1
4. Which of the following are examples of permutations?a. Arranging 5 people on a bench that seats 5
b. Choosing 4 letters from 6 and writing all possible arrangements of the choices
c. Selecting a committee of 3 from 10 women
Understanding Break
1. What is the short way of writing (6)(5)(4)(3)(2)(1)?6!
2. Is (2!)(3!) the same as 6! ?
3. Which of the following is the value of 0! ?a. 0 b. 1 c. -1 d. ±1
4. Which of the following are examples of permutations?a. Arranging 5 people on a bench that seats 5
b. Choosing 4 letters from 6 and writing all possible arrangements of the choices
c. Selecting a committee of 3 from 10 women
Understanding Break
1. What is the short way of writing (6)(5)(4)(3)(2)(1)?6!
2. Is (2!)(3!) the same as 6! ?No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1)
3. Which of the following is the value of 0! ?a. 0 b. 1 c. -1 d. ±1
4. Which of the following are examples of permutations?a. Arranging 5 people on a bench that seats 5
b. Choosing 4 letters from 6 and writing all possible arrangements of the choices
c. Selecting a committee of 3 from 10 women
Understanding Break
1. What is the short way of writing (6)(5)(4)(3)(2)(1)?6!
2. Is (2!)(3!) the same as 6! ?No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1)
3. Which of the following is the value of 0! ?a. 0 b. 1 c. -1 d. ±1
4. Which of the following are examples of permutations?a. Arranging 5 people on a bench that seats 5
b. Choosing 4 letters from 6 and writing all possible arrangements of the choices
c. Selecting a committee of 3 from 10 women
Understanding Break
1. What is the short way of writing (6)(5)(4)(3)(2)(1)?6!
2. Is (2!)(3!) the same as 6! ?No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1)
3. Which of the following is the value of 0! ?a. 0 b. 1 c. -1 d. ±1
4. Which of the following are examples of permutations?a. Arranging 5 people on a bench that seats 5
b. Choosing 4 letters from 6 and writing all possible arrangements of the choices
c. Selecting a committee of 3 from 10 women
Understanding Break
1. What is the short way of writing (6)(5)(4)(3)(2)(1)?6!
2. Is (2!)(3!) the same as 6! ?No; (2)(1)(3)(2)(1) ≠ (6)(5)(4)(3)(2)(1)
3. Which of the following is the value of 0! ?a. 0 b. 1 c. -1 d. ±1
4. Which of the following are examples of permutations?a. Arranging 5 people on a bench that seats 5
b. Choosing 4 letters from 6 and writing all possible arrangements of the choices
c. Selecting a committee of 3 from 10 women
Homework
Homework
"The inner fire is the most important thing mankind possesses."- Edith Sodergran
p. 174 #1-25 odd