combinatorics class booklet topic 2 solutions

7/30/2019 Combinatorics Class Booklet Topic 2 Solutions

1/8

1. A university science program requires that students take five courses in their first year. All students must takeChemistry and Biology as two of their courses, and then must choose from:

- One of either Physics, Botany, or Astronomy- One of Calculus, Linear Algebra, Number Theory, or Stascs-

One of Psychology, Sociology, Economics, Philosophy, or Anthropology

How many schedule opons are possible for the first year

2. A bank card pass-code must consist of any four digits.(a) Provide a number for each blank to determine the total possible bank card pass-codes.

(b) Determine the number of bank card pass-codes possible if no digit can repeat.

(c) no digit can repeat and the first digit cannot be 0.

3. In a parcular urisdicon, license plates consist of any three leers (where the first of which is not the leersI or O), followed by any three non-repeangdigits (the first of which cannot be 1 or 0). Determine

the total number of possible license plates.

4. The final score of a hockey game was Jets 4, Flames 2. How many game scores were possible at the end of thefirst period

# of opons for first digit for 2nd

digit for third digit

=

5.

gchoicesg

gchoicesg

gchoicesg

3 4 5 = gpon

Mustgtakeg/gogchoiceghere.ggSogdinregard.gg(cosidergarranmetsgforgthegrest)g

10

Fourth

10 10 10

Any digit except 0:

1, 2, 3, 4, 5, 6, 7, 8, or 9

(9 opons)

= g

Catgbegzerog Cant be first digit

9 9 8 7 = g

10 9 8 = g 7


2/8

7. Consider the word JACKSON. How many ways are there to(a) Arrange all of the leers in the word

8. A school play has three male parts. The drama club has 7 boys.(a) Determine the number of arrangements of boys for the three parts, using

the fundamental counng principal

(b) Determine the number of arrangements of boys, using the formula

(c) Determine the number of arrangements of boys possible if Wagner has been promised the lead role.

(d) Determine the number of arrangements if Wagner has been promised a role.

(b) Arrange all of the leers in the word if thefirst two leers must be JA (igthatgorder)

(d) Arrange all of the leers in the word if theleer must be a consonant

(c) Make an arrangement using any two leers.

=!

( )!

6.
http://www.google.ca/imgres?imgurl=http://www.marywinspear.ca/uploads/images/rooms/charlie_White_theatre_05.jpg&imgrefurl=http://www.marywinspear.ca/index.php?centre=Live-Event-Theatre&h=338&w=450&sz=36&tbnid=Hv8BOmf2syjKGM:&tbnh=91&tbnw=121&zoom=1&usg=__DLOF24r4Zl44xZ_UREnZtysI-O4=&docid=foFyrTeGy37x1M&hl=en&sa=X&ei=UV3DUKDjJrDxiQKM9ID4Bg&ved=0CDwQ9QEwBA&dur=624


3/8

9. Determine the number of different arrangements using all the leers of the word ACCESSES that

10.A co-ed basketball team, consisng of 4 boys and 3 girls, is lining up for a photo.(a) How many different arrangements of the team are possible

(b) How many arrangements are possible if all of the boys are on the le side,girls on the right side

(c) How many arrangements are possible if the boys and girls must alternate

(d) How many arrangements are possible if the boys all must be kept together

Permutaons (arrangements) problems where certain obects must be kept tgether

Count the obects that must

be kept together as one unit

Sluo

Number of permutaons, counng

the together obects as one

Number of permutaons of the

together obects among themselves

11.

(a) begin with exactly two Ss.

(b) begin with at least two Ss.

(c) Explain why the answers in quesons (a) and (b) are different.


4/8

Factorial notaon expressions can be simplified / canceled out.

6!

3!=

6543!

3!= 6 5 4gorgg120

13.Without using a calculator, simplify each:(a) 9!

7! (b)

8!

5!3!

14. Simplify the expression: (+2)!!

15.A catalogue designer for a ewelry store is construcng an ad to promote a sale on ladies watches. The ad isto have two of the on-sale watches displayed one on the top of the page, and one on the boom. Aer

showing the store manager all possibilies for the page, 30 different ad opons were given.

How many models of ladies watches does the store have on sale

12.

2

= 20

2

2 0 = 0

( 5)( + 4) = 20

We can also nlve ao equao

2 = 20 !

(2)!= 20

(1)(2)!

(2)!

= 20

= or =

Gal Cancel

the factorials!

Strategy: Given an expression !, expand it out

unl you can cancel the factorial terms out.

extraneous g


5/8

Prmuaisisiobeswtsri

A grocery store manager is making a fruit arrangement in a display window. (Could happen) He has

an apple, an orange, and two banans four items of fruit in total.

How many arrangements are possible

Would the display be different if he changed the posion of the last two fruits

1. Below is a list of all possible arrangements of the leers A, O, B1, B2. Crisiusaysrdudas(rad)srw that is, entries that representan idencal fruit arrangement.

2. Hismaysdwesarranm are there of four fruits, consisng of 1 apple, 1 orange, and 2bananas

3. How many disnct arrangements are there if thefirisruwmusosoaaa

4. How many disnct arrangements are there if thefirsruwsmusosasoaaa

5. Suppose we replaced each orange with a banana. (crazy!) How many dwesarranmsiuldstrsost

6. How does the number of disnct arrangements ofobects relate to the number ofrepeons

,1,,2

,2,, 1,1,2,

,2,1,

, , 1,2

, , 2,1

2, 1 ,,

2, ,, 12, 1 ,,

2, ,1,

2, , 1,

2, , , 1

,1,, 2

,2,, 1,1, 2,

,2,1,

, , 1, 2

, , 2, 1

1,2 ,,

1,2 ,,1,, 2,

1,,2,

1, , , 2

1, , , 2

Must be bananas!

Must be a banana!

, 1,3, 2

, 2, 3, 1

, 1,2, 3

, 2, 1, 3

, 3, 1,2

, 3, 2,1

1,2,, 3

1,2, 3,

1,3, 2,

1,,2, 3

1, 3, , 2

1, , 3, 2

2,1,, 3

2,3,, 1

2,1, 3,

2,3, 1,

2, , 1, 3

2, , 3, 1

3, 1,, 2

3, 2,, 1

3, 1,2,

3, 2, 1,

3, , 1, 2

3, , 2, 1


6/8


7/8


8/8

combinatorics class booklet topic 2 solutions

Documents