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

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

Permutations

Mathematics 4

February 22, 2012

Mathematics 4

Permutations

Definition of permutations

Permutation

A permutation is an arrangement of all or part of a set of objects

Mathematics 4

Permutations

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

Example 1

How many three-letter ”words” can you form from the wordCHEMISTRY?

Mathematics 4

Permutations

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

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

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

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

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

Example 4

How many ways can you arrange the letters in the wordMATHEMATICS?

Mathematics 4

Permutations

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

Permutation formulas

Circular Permutations

The number of permutations of n distinct objects arranged in acircle is

(n− 1)! (3)

Mathematics 4

Permutations

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

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

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

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

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

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

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

Definition of combinations

Combination

A combination is selecting objects with no regard to order

Mathematics 4

Permutations

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

Example 8

How many teams of 3 students from your class be selected for theteam category of the Math intersection?

Mathematics 4

Permutations

Example 9

How many subsets of a set of 10 elements have either 3 or 4elements?

Mathematics 4

Permutations

Example 10

How many different committees of 4 to 6 persons be chosen froma society of 28 members?

Mathematics 4

Permutations

Example 11

How many ways can a team of 3 boys and 4 girls be chosen fromyour class?

Mathematics 4

Permutations