Name: Version #1Instructor: Annette McP

Math 10120 Exam 1Sept. 19, 2019.

• The Honor Code is in e↵ect for this examination. All work is to be your own.• Please turn o↵ all cellphones and electronic devices.• Calculators are allowed.• The exam lasts for 1 hour and 15 minutes.• Be sure that your name and your instructor’s name are on the front page of your exam.• Be sure that you have all 11 pages of the test.

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!

1 . (•) (b) (c) (d) (e)

2 (•) (b) (c) (d) (e)

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3. (•) (b) (c) (d) (e)

4. (•) (b) (c) (d) (e)

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5. (•) (b) (c) (d) (e)

6. (•) (b) (c) (d) (e)

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7. (•) (b) (c) (d) (e)

8. (•) (b) (c) (d) (e)

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9. (•) (b) (c) (d) (e)

10. (•) (b) (c) (d) (e)

Please do NOT write in this box.

Multiple Choice

11.

12.

13.

14.

15.

Total

Name:

Instructor:

Math 10120 Exam 1Sept. 19, 2019.

• The Honor Code is in e↵ect for this examination. All work is to be your own.• Please turn o↵ all cellphones and electronic devices.• Calculators are allowed.• The exam lasts for 1 hour and 15 minutes.• Be sure that your name and your instructor’s name are on the front page of your exam.• Be sure that you have all 11 pages of the test.

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!

1. (a) (b) (c) (d) (e)

2 (a) (b) (c) (d) (e)

........................................................................................................................

3. (a) (b) (c) (d) (e)

4. (a) (b) (c) (d) (e)

........................................................................................................................

5. (a) (b) (c) (d) (e)

6. (a) (b) (c) (d) (e)

........................................................................................................................

7. (a) (b) (c) (d) (e)

8. (a) (b) (c) (d) (e)

........................................................................................................................

9. (a) (b) (c) (d) (e)

10. (a) (b) (c) (d) (e)

Please do NOT write in this box.

Multiple Choice

11.

12.

13.

14.

15.

Total

TEEN ANTTENANT E

2. Initials:

Multiple Choice

1.(5pts) LetU = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 3, 4, 5, 6}B = {2, 4, 6, 8, 10}.C = {4, 5, 6, 7, 8}.

Which of the following sets is equal to (A \B) [ C0 ?

(a) {1, 2, 3, 4, 6, 9, 10} (b) {1, 2, 3, 9, 10} (c) {2, 4, 5, 6, 7, 8}

(d) {2, 3} (e) {2}

2.(5pts) Consider the following sets

U = {English Words}

A = {Six letter English words}

B = {English words with at least two di↵erent (see below) vowels}

C = {English words starting with the letter “f”}

Which one of the following words is in the set

(A0 [ B) \ C

[Note: two vowels are di↵erent if they are di↵erent letters of the alphabet. For example theword “fee” is not in the set B but the word “fie” is in the set B]

(a) four (b) table (c) cat

(d) fright (e) cable

An B 2,4 6

C 1 2,3 9,10An B U C 1 2,3 4 6 9,10

X EA HC EA f CEIA BEC EA 4 C

f our C C since it STARTS WITH ff our E B sirree it HAS 2 vowelsso four C A u B AND four E C

Hence Dfourt AUB AC

3. Initials:

3.(5pts) In a survey of 70 students who used the sports facilities at Notre Dame, 30 usedthe climbing wall facilities regularly, 40 used the swimming facilities regularly and 10 usedneither the climbing wall nor the swimming facilities on a regular basis. How many studentsused both the climbing wall facilities and the swimming facilities regularly?

(a) 10 (b) 20 (c) 15 (d) 70 (e) 5

4.(5pts) Matt likes to order a burger from “Paul’s Famous Hamburgers” on Friday night.Matt must first choose between a small or a large hamburger patty. He must then choosesome subset of toppings (he is free to choose no toppings or all toppings or some other subsetof toppings) from the following list:

lettuce tomato raw onionbacon cheese fried egg

pineapple beetroot cooked onions

Matt must then choose at most two (i.e. zero, one or two) of the following sauces:

BBQ mustard sweet chilihot chili mayonnaise tomato saucePaul’s special sauce

How many di↵erent orders can Matt place?

(a) 2 · 29 · 29 (b) 2 · 9 · 7 (c) 2 · 9! · P (7, 2)

(d) 2 · 29 · C(7, 2) (e) 2 · 29 · 27

u C

s

NCC 30 n Sn cn s 40 IN EXM Csu c 7 10 nlsue n1s tnLc Nlsnc

n Sue nlu n Csucn U 70to 10 60

60 30 40 NHC

9 Toppings

7 sauces

29

9 I it 7 13Q 2 Cft o tcC7 1 47,2choices

PATTY Toppings Scene.la orzstep i step 2 Steps Mutt principle

4. Initials:

5.(5pts) How many di↵erent words, including nonsense words can be made by rearrangingthe letters of the word

KERFUFFLE

(a)9!

2! · 3! (b) 9! (c)6!

2! · 3!

(d)P (9, 6)

2! · 3! (e) P (9, 6)

6.(5pts) The University of Wattamolla creates ID numbers for its students consisting of 6digits. Each ID number must start and end with an odd digit. How many di↵erent such IDnumbers are possible?

(a) 250, 000 (b) 200, 000 (c) 1, 000, 000

(d) 55 · 4 (e) 20 · P (5, 4)

K l

9 ways to permute letters E 2

2731 ways to permuteR 1

Repeated Letters F 3u lL I

Malt principle5 to to go go 5 10Digits

ODD ODD 5 are odd

25 X 10,000 250,000

5. Initials:

7.(5pts) Mogo Zoo has a list of 10 docents who volunteer to act as guides during specialevents. The zoo sent out an e-mail calling for at least 8 volunteers for the annual MarsupialMania Festival. How many di↵erent subsets of the list of docents have at least 8 elements.

(a) 56 (b) 45 (c) 720

(d) 820 (e) 1

8.(5pts) The grid shown below is a street map of Gameville. In how many ways can a taxiget from the Airport at A to the Ball game at B if the cab always travels south or east anddoes not pass through the intersection at C ?

S

EA

B

C

(a) C(10, 4)� C(4, 2)C(6, 2) (b) C(10, 4)

(c) C(4, 2)C(6, 2) (d) C(4, 2) + C(6, 2)

(e) C(10, 4)� [C(4, 2) + C(6, 2)]

Choi subets of a set with co elements

hath are ntle.noement

8 or 9 or 10 elements56 Cho 81 Clio 9 tcl 10,10

EFFEE E Es

ss Routes THATDONOTSPASS THRU C

TOTAL Routes Routes Thru CC A 7C f c B

Cao 4 g 4 2

Clio 4 CC4,2 CC6,47

6. Initials:

9.(5pts) A student organization has 10 members from Fisher Hall, 15 members from BadinHall and 22 members from Walsh hall. The organization must select three members fora committee. In how many ways can the selection be made so that all three committeemembers are from the same hall?

(a) C(10, 3) + C(15, 3) + C(22, 3) (b) C(10, 3) · C(15, 3) · C(22, 3)

(c) P (10, 3) · P (15, 3) · P (22, 3) (d) P (10, 3) + P (15, 3) + P (22, 3)

(e)C(47, 3)

3!

10.(5pts) A bag contains 8 red marbles (numbered 1-8) and 5 green marbles (numbered 1-5).

How many di↵erent samples of 5 marbles, with 2 red marbles and 3 green marbles, can bedrawn from the bag?

(a) C(8, 2) · C(5, 3) (b) P (8, 2) · P (5, 3)

(c) C(8, 2) + C(5, 3) (d) P (8, 2) + P (5, 3)

(e)C(13, 5)

2! · 3!

select 3Either 3fromFisher 410,3

i or t

3 FaronBABIN CG5,3or t

3 from Walsh 22,3

x

t T s Istep 1 step2choose choose2Rect 3 Green

7. Initials:

Partial Credit

11.(12pts) A survey of 100 visitors to the campus of The University of Not The Same wereasked which of three popular activities they did when on campus. The results of the surveywere as follows:50 of them (The set F in the picture) went to see the university football team, The FightingThai Fish, play a game on campus.30 of them (The set G in the picture) went to see The Golden Gnome,and 25 of them (The set T in the picture) took photos in front of Touchdown Venus.

Ten of those surveyed went to see the football team play a game and went to see theGolden Gnome,7 of them went to see the football team play and took photos in front of Touchdown Venus,and 12 of them went to see the golden gnome and took photos in front of Touchdown Venus.Three of those surveyed did all three activities.

(a) Place the appropriate number of ele-ments in each of the 8 basic regions in thediagram on the right.

G = Went to see The Golden Gnome.T = Took photos in front of Touchdown Venus.

F = Went to see The Fighting Thai Fish play.

U

T

GF

(b) Shade the region(s) in the diagram onthe right which correspond(s) to visitors sur-veyed who visited exactly two of the threeactivities.

G = Went to see The Golden Gnome.T = Took photos in front of Touchdown Venus.

F = Went to see The Fighting Thai Fish play.

U

T

GF

(c) How many of the visitors surveyed did not do any of the three activities mentioned?

100

36 7 113

F 50 4 9

E 3 9 II79

F NG 10FAT 7GAT I 12

F NGATI 3

E

D

8. Initials:

For Questions 12-14, you may express your answers using the notation forpermutations, combinations, powers and factorials, where appropriate

12.(12pts) The rules of the Daingean lottery are as follows:

• You select a subset (unordered) of 4 di↵erent numbers from the numbers 1- 50.• The lottery committee also chooses a subset (unordered) of four numbers from thenumbers 1-50, to be designated as winning numbers.

• You will win a prize if you have at least 3 of the four winning numbers in yourselection (order does not matter).

(a) How many di↵erent (unordered) subsets of 4 (di↵erent) numbers can be chosen fromthe numbers 1-50 ?

For parts (b) and (c), assume that four of the numbers have been designated as“winning numbers” and the other 46 have been designated as “non-winning numbers”.

(b) How many subsets from part (a) have exactly 3 “winning numbers” in them?

(c) How many subsets from part (a) have at least 3 “winning numbers” in them?

50,4

1184J

L yTo create such a sorbet4,31 446,2744

Step t step 2Choose 3fromThe4 W choose 1 N w

at Least 3 W'is Ex 3 w s or Ex 4 w s

184 t CCGs4

44131.46 t 1

9. Initials:

13.(12pts) The Smite museum has 20 paintings of lines and primary colored rectangles createdby Mr. Mundrain, and 10 paintings of flowers by Georgia O’Leaf.

(a) The museum wishes to select 5 paintings from the 20 by Mr. Mundrain, and 4 paintingsfrom the 10 by Georgia O’Leaf in their possession, for an exhibition. In how many di↵erentways can they select the nine paintings for their exhibition?

Assume now (for parts (b) and (c)) that a selection of 5 paintings by Mr. Mundrain and 4by Georgia O’leaf has been made for the exhibition. All 9 of the paintings chosen will bearranged in a row.

(b) How many such arrangements can be made if the paintings are placed so that a paintingof Georgia O’Leaf appears at both ends of the row?

(c) How many arrangements of the 9 paintings can be made if the paintings are placed sothat a painting of Mr. Mundrain appears at both ends of the row and the paintings alternatebetween paintings of Mr. Mundrain and Ms. O’Leaf?

i.e. an arrangement of the form M O M O M O M O M where an “M” denotes a paintingby Mundrain and an “O” denotes a painting by O’Leaf.

Ess.eestep 1 Step 25 by Mr M 4 byGok

4 I b E 4 3 2 1 127Ok

Step 1 steps 3 9 arrange rest steps

First Arrange 5M's 5Hult prime Next n 40 s 4

In all 5 4 ways

10. Initials:

14.(12pts) (a) Let A be the set of 4 letter words, including nonsense words, that can be madefrom the letters of the word

F I G H T I N G,

if letters cannot be repeated. (e.g. FIGH is considered a word, but FIIH is not). How manyelements are in the set A?

(b) Let B be the set of 4 letter words, including nonsense words, that can be made from theletters of the word

I R I S H,

if letters can be repeated. (e.g. IIII is considered a word as is RRRR). How many elementsare in the set B?

(c) What is A \ B? (Justify your answer)

(d) How many elements does the set A [B have?

I 6 Letters Available

6 I 4 3 Pcb 4

I Letters Available

4 4 4 4 44

A womb in AAB CANNOT HAVERepeated Letters because it is in A Since it isin B AND Does Not Have Repeated Letters it has the Letters1 R S and H in it However THER is no such word in A

so A NB 10

n Av B nl Alin 1Bsince A and B are nut Exclusive

6 5 4.3 44360 t 256 16162

11. Initials:

15.(2pts) You will be awarded these two points if you write your name in CAPITALS and youmark your answers on the front page with an X (not an O) . You may also use this page for

ROUGH WORK