permutations
DESCRIPTION
TRANSCRIPT
![Page 1: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/1.jpg)
Quiz 6
You may answer each item as a product of factors (e.g. 4 · 3 · 2)
1 How many 4-digit whole numbers can be formed if there is norepetition of digits?
2 How many 4-digit odd whole numbers can be formed ifrepetition of digits is allowed?
3 How many whole numbers less than 10000 can be formedusing odd digits?
4 How many 5-digit whole numbers can be formed if odd andeven digits should alternate and there is no repetition ofdigits?
5 How many 2-digit whole numbers can be formed that are notperfect squares?
Mathematics 4
Permutations
![Page 2: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/2.jpg)
Quiz 7
You may answer each item as a product of factors (e.g. 4 · 3 · 2)
1 How many ways can you arrange the letters in the wordCESIUM?
2 In a class of 15 students, how many ways can a president, vicepresident, and a secretary be chosen?
3 How many ways can 10 people be lined up if 4 of them alwayshave to be together?
4 How many ways can 5 prom couples be lined up if each couplemust be together?
5 How many ways can the same 5 prom couples be arranged ina line if each girl is between her date and another girl, orbeside her date only?
Mathematics 4
Permutations
![Page 3: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/3.jpg)
Permutations
Mathematics 4
February 22, 2012
Mathematics 4
Permutations
![Page 4: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/4.jpg)
Definition of permutations
Permutation
A permutation is an arrangement of all or part of a set of objects
Mathematics 4
Permutations
![Page 5: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/5.jpg)
Permutation formulas
Linear Permutation
The number of permutations of n distinct objects taken r at atime is
nPr =n!
(n− r)!(1)
Mathematics 4
Permutations
![Page 6: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/6.jpg)
Example 1
How many three-letter ”words” can you form from the wordCHEMISTRY?
Mathematics 4
Permutations
![Page 7: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/7.jpg)
Example 2
In how many ways can 8 people line up to get on a bus:
a. if three specific persons insist on following each other?
b. if two specific persons refuse to follow each other?
Mathematics 4
Permutations
![Page 8: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/8.jpg)
Example 2
In how many ways can 8 people line up to get on a bus:
a. if three specific persons insist on following each other?b. if two specific persons refuse to follow each other?
Mathematics 4
Permutations
![Page 9: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/9.jpg)
Example 3
How many five-letter ”words” can you form from the letters A, E,I, O, U, B, C, D, F, G, H, J, with no repetition of letters:
a. if vowels and consonants must alternate?
b. if each word must start and end with a vowel?
Mathematics 4
Permutations
![Page 10: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/10.jpg)
Example 3
How many five-letter ”words” can you form from the letters A, E,I, O, U, B, C, D, F, G, H, J, with no repetition of letters:
a. if vowels and consonants must alternate?b. if each word must start and end with a vowel?
Mathematics 4
Permutations
![Page 11: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/11.jpg)
Permutation formulas
Distinguishable Permutations
The number of permutations of n things of which n1 are of onekind, n2 of a second kind, ..., nk of a kth kind is
n!
n1!n2!...nk!(2)
Mathematics 4
Permutations
![Page 12: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/12.jpg)
Example 4
How many ways can you arrange the letters in the wordMATHEMATICS?
Mathematics 4
Permutations
![Page 13: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/13.jpg)
Example 5
How many ways can 3 rose bushes, 4 santan bushes, and 2gumamela bushes be arranged along a property line if one does notdistinguish between bushes of the same kind?
Mathematics 4
Permutations
![Page 14: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/14.jpg)
Permutation formulas
Circular Permutations
The number of permutations of n distinct objects arranged in acircle is
(n− 1)! (3)
Mathematics 4
Permutations
![Page 15: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/15.jpg)
Example 6
How many ways can 5 couples be seated around a table
a. if there are no restrictions?
b. if each couple is to seat together?c. if each couple is to seat across each other?d. if men and women alternate?
Mathematics 4
Permutations
![Page 16: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/16.jpg)
Example 6
How many ways can 5 couples be seated around a table
a. if there are no restrictions?b. if each couple is to seat together?
c. if each couple is to seat across each other?d. if men and women alternate?
Mathematics 4
Permutations
![Page 17: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/17.jpg)
Example 6
How many ways can 5 couples be seated around a table
a. if there are no restrictions?b. if each couple is to seat together?c. if each couple is to seat across each other?
d. if men and women alternate?
Mathematics 4
Permutations
![Page 18: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/18.jpg)
Example 6
How many ways can 5 couples be seated around a table
a. if there are no restrictions?b. if each couple is to seat together?c. if each couple is to seat across each other?d. if men and women alternate?
Mathematics 4
Permutations
![Page 19: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/19.jpg)
Permutation formulas
Ring Permutations
The number of permutations of n distinct objects arranged in acircle with no distinction for clockwise or counterclockwise is
(n− 1)!
2(4)
Mathematics 4
Permutations
![Page 20: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/20.jpg)
Example 7
How many ways can 7 keys be arranged on a key ring
a. if there are no restrictions?
b. if two keys must always be together?
Mathematics 4
Permutations
![Page 21: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/21.jpg)
Example 7
How many ways can 7 keys be arranged on a key ring
a. if there are no restrictions?b. if two keys must always be together?
Mathematics 4
Permutations
![Page 22: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/22.jpg)
Definition of combinations
Combination
A combination is selecting objects with no regard to order
Mathematics 4
Permutations
![Page 23: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/23.jpg)
Combination formula
Combination
The number of combinations of n distinct objects taken r at atime is
nCr =nPr
r!=
n!
(n− r)!r!(5)
Mathematics 4
Permutations
![Page 24: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/24.jpg)
Example 8
How many teams of 3 students from your class be selected for theteam category of the Math intersection?
Mathematics 4
Permutations
![Page 25: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/25.jpg)
Example 9
How many subsets of a set of 10 elements have either 3 or 4elements?
Mathematics 4
Permutations
![Page 26: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/26.jpg)
Example 10
How many different committees of 4 to 6 persons be chosen froma society of 28 members?
Mathematics 4
Permutations
![Page 27: Permutations](https://reader031.vdocuments.site/reader031/viewer/2022020217/54536690af795940458b4bbc/html5/thumbnails/27.jpg)
Example 11
How many ways can a team of 3 boys and 4 girls be chosen fromyour class?
Mathematics 4
Permutations