1. The document contains examples and formulas for permutations and combinations. It discusses linear permutations, distinguishable permutations, circular permutations, and ring permutations. 2. One example asks how many five-letter words can be formed from a set of vowels and consonants if they must alternate with no repetition. 3. The combination formula is presented, and examples ask how to select teams and committees from groups of people.

1. 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 no
repetition of digits?
2 How many 4-digit odd whole numbers can be formed if
repetition of digits is allowed?
3 How many whole numbers less than 10000 can be formed
using odd digits?
4 How many 5-digit whole numbers can be formed if odd and
even digits should alternate and there is no repetition of
digits?
5 How many 2-digit whole numbers can be formed that are not
perfect squares?
Mathematics 4
Permutations

2. 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 word
CESIUM?
2 In a class of 15 students, how many ways can a president, vice
president, and a secretary be chosen?
3 How many ways can 10 people be lined up if 4 of them always
have to be together?
4 How many ways can 5 prom couples be lined up if each couple
must be together?
5 How many ways can the same 5 prom couples be arranged in
a line if each girl is between her date and another girl, or
beside her date only?
Mathematics 4
Permutations

3. Permutations
Mathematics 4
February 22, 2012
Mathematics 4
Permutations

4. Definition of permutations
Permutation
A permutation is an arrangement of all or part of a set of objects
Mathematics 4
Permutations

5. Permutation formulas
Linear Permutation
The number of permutations of n distinct objects taken r at a
time is
n!
n Pr = (1)
(n − r)!
Mathematics 4
Permutations

6. Example 1
How many three-letter ”words” can you form from the word
CHEMISTRY?
Mathematics 4
Permutations

7. 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?
Mathematics 4
Permutations

8. 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

9. 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?
Mathematics 4
Permutations

10. 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

11. Permutation formulas
Distinguishable Permutations
The number of permutations of n things of which n1 are of one
kind, n2 of a second kind, ..., nk of a kth kind is
n!
(2)
n1 !n2 !...nk !
Mathematics 4
Permutations

12. Example 4
How many ways can you arrange the letters in the word
MATHEMATICS?
Mathematics 4
Permutations

13. Example 5
How many ways can 3 rose bushes, 4 santan bushes, and 2
gumamela bushes be arranged along a property line if one does not
distinguish between bushes of the same kind?
Mathematics 4
Permutations

14. Permutation formulas
Circular Permutations
The number of permutations of n distinct objects arranged in a
circle is
(n − 1)! (3)
Mathematics 4
Permutations

15. Example 6
How many ways can 5 couples be seated around a table
a. if there are no restrictions?
Mathematics 4
Permutations

16. 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?
Mathematics 4
Permutations

17. 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?
Mathematics 4
Permutations

18. 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

19. Permutation formulas
Ring Permutations
The number of permutations of n distinct objects arranged in a
circle with no distinction for clockwise or counterclockwise is
(n − 1)!
(4)
2
Mathematics 4
Permutations

20. Example 7
How many ways can 7 keys be arranged on a key ring
a. if there are no restrictions?
Mathematics 4
Permutations

21. 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

22. Definition of combinations
Combination
A combination is selecting objects with no regard to order
Mathematics 4
Permutations

23. Combination formula
Combination
The number of combinations of n distinct objects taken r at a
time is
n Pr n!
n Cr = = (5)
r! (n − r)!r!
Mathematics 4
Permutations

24. Example 8
How many teams of 3 students from your class be selected for the
team category of the Math intersection?
Mathematics 4
Permutations

25. Example 9
How many subsets of a set of 10 elements have either 3 or 4
elements?
Mathematics 4
Permutations

26. Example 10
How many different committees of 4 to 6 persons be chosen from
a society of 28 members?
Mathematics 4
Permutations

27. Example 11
How many ways can a team of 3 boys and 4 girls be chosen from
your class?
Mathematics 4
Permutations