Counting Unit

Review Sheet

1. There are five choices of ice cream AND three choices of cookies. a) How many different

desserts are there if you have one scoop of ice cream AND one cookie?

_________ • ________

Ice cream cookie

5 • 3____

Ice cream cookie

15

b) How many different desserts are there if you have either one scoop of ice cream OR a cookie?

5(ice cream) + 3(cookie)

8

2. How many different 3-letter “words” can be formed from the

letters in the word CANOE? • • ____

5 • 4 • 3___

60

3. How many different ways can 5 children arrange themselves for a game of ring-around-the- rosie?(5 – 1)!

4!

4 • 3 • 2 • 1 =

24

4. How many different ways can a teacher choose 10 homework

problems from a set of 25?

25C10

25! =

(25-10)! 10!

25•24•23•22•21•20•19•18•17•16•15!

15! 10•9•8•7•6•5•4•3•2•1

3,268,760

5. How many different arrangements are there of the digits 166555? 6! =

2! 3!

6 • 5 • 4 • 3!

2 • 3!

60

6. A child has 10 identically shaped blocks – 4 red, 3 green, 2 yellow, and 1 blue. How many different stacks of all 10 blocks are possible?

10! =

4! 3! 2! 1!

10 • 9 • 8 •7 • 6 • 5 • 4!

4! 3 • 2 • 2

151,200

12

12,600

7. How many ways can 10 people be seated around a circular table if the host and hostess cannot be seated together?(10 – 1)! = 362,880 If the host and hostess do sit together, they would

be counted as one, so now it would be asking for 9

people seated in a circle.

(9 – 1)! = 8! = 40320So to find the ways they do not sit together, subtract

the two answers

9! – 8! = 362,880 – 40320 =

322,560

8. A committee of 4 is to be chosen from a club with 10 male and 12 female members. If at least 2 women must be chosen how many ways can this be done

____ • ___

female male

12 C2 • 10C2

12 C2 10C2 + 12C3 10 C1 +12C4 10 C0

13. Find the number of arrangements of the word LEVELED

7! =

3! 2!

7 • 6 • 5 • 4 • 3!

3! 2

840

2

420

14. How many 4 digit numbers can be made using the digits 0, 1, 2, 3, 4, 5 if repetition is not allowed?_ • _ • _ • _ =

5 • 5 • 4 • 3 =

300

(How many 4 digit numbers can be made using the digits 0, 1, 2, 3, 4, 5 ifrepetition is not allowed?) 15. How many of them are odd?

_ • _ • _ • _ =

4 • 4 • 3 • 3

144

(How many 4 digit numbers can be made using the digits 0, 1, 2, 3, 4, 5 if repetition is not allowed?) 16. Do #14 if repetition is allowed

_ • _ • _ • _ =

5 • 6 • 6 • 6

1080

17. How many ways can you answer a 15-question always-sometimes-never geometry quiz

• • • • • • • • • • • • • •__

3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3

14348907

3. Suppose you take 4 different routes to Trenton, the 3 different routes to Philadelphia.

How many different

routes can you take

for the trip to

Philadelphia by way

of Trenton?

________ • _________

Trenton Philadelphia

___4____ • ___3_____

12

4. You have 10 pairs of pants, 6 shirts, and 3 jackets.

How many outfits

can you have

consisting of a

shirt, a pair of

pants, and a

jacket?

______•______•______

Shirts Pants Jackets

___6__•__10__•__3___

180

5. Fifteen people line up for concert tickets.

a) How many

different

arrangements are

possible?

__•__•__•__•__•__•__•__•__

•__•__•__•__•__• _=

15•14•13•12•11•10•9•8•7•6•5•4•3•2•1 =

1,307,674,368,000

b) Suppose that a

certain person must

be first and another

person must be last.

How many

arrangements are now

possible?

1 •__•__•__•__•__•__•__•__

•__•__•__•__•__• 1 =

1•13•12•11•10•9•8•7•6•5•4•3•2•1•1 =

6,227,020,800

6) Using the letters A, B, C, D, E, Fa) How many “words”can be made using all 6letters?6 • 5 • 4 • 3 • 2 • 1 = 720b) How many of thesewords begin with E ?1 • 5 • 4 • 3 • 2 • 1 = 120c) How many of thesewords do NOT beginwith E? 720 –120 = 600d) How many 4-letterwords can be made ifno repetition is allowed?6•5•4•3 = 360

e) How many 3-letterwords can be made ifrepetition is allowed?6 • 6 • 6 = 216f) How many 2 OR 3letter words can bemade if repetition isnot allowed? 6•5+6•5•4 = 30 + 120 = 150g) If no repetition isallowed, how manywords containing atleast 5 letters can bemade? (both letter 6a)720 + 720 = 1440

6) Using the letters A, B, C, D, E, Fa) How many “words”

can be made using all 6

letters?

6P6 = 6 • 5 • 4 • 3 • 2 • 1 = 720

b) How many of these

words begin with E ?

1 • 5 • 4 • 3 • 2 • 1 = 120

c) How many of these

words do NOT begin

with E? 720 –120 = 600

d) How many 4-letter

words can be made if

no repetition is allowed?

6P4 = 6•5•4•3 = 360

e) How many 3-letter

words can be made if

repetition is allowed?

6 • 6 • 6 = 216

f) How many 2 OR 3 letter

words can be made if

repetition is not allowed?

6P2 + 6P3 =

6•5 + 6•5•4 = 30 + 120 = 150

g) If no repetition is allowed,

how many words containing

at least 5 letters can be made

6P5 + 6P6 =

720 + 720 = 1440

7. How many distinguishable permutations can be made using all the letters of:

a) GREAT

__•__•__•__•__

5 • 4 • 3 • 2 • 1

5!

120

b) FOOD

4!

2!

4 • 3 • 2!

2!

12

c) TENNESSEE

9!_________

4! 2! 2!1!

9 • 8 • 7 • 6 • 5 • 4!

4! 2 • 2

15,120

4

3,780

8. Suppose you have 3 red flags, 5 green flags, 2 yellow flags, and 1 white flag. Using all the flags in a row, how many distinguishable signals can be sent?

11! =

3! 5! 2!1!

11 • 10 • 9 • 8 • 7 • 6 • 5! =

3 • 2 • 5! • 2

332,640 =

12

27,720

9. How many ways can 7 people be seated in a circle?

(7-1)! =

720

10. If you have a dozen different flowers and wish to arrange them so there is one in the center and the rest in a circle around them, how many arrangements are possible? 12 • (11-1)! =

Center Circle

12 • 3,628,800 =

43,545,600

11. Note: zero can never be the first digit of a “__-digit number”.

a) How many 4-

digit numbers

contain no nines?

__ • __ • __ • __ 8 • 9 • 9 • 9 =

5832

b) How many 4-

digit numbers contain

AT LEAST ONE nine?

__ • __ • __ • __ 9 • 10 • 10 • 10 –

8 • 9 • 9 • 9 =

9000 – 5832 =

3168

12. How many 10-letter words can you make if no letter can be repeated?

Set up using the

fundamental counting

principle.

__ • __ • __ • __ • __ •__

• __ • __ • __ • __

26•25•24•23•22•21•20•

19•18•17 =

1,927,522,397,000

Then using

permutation notation

26 P10 =

26! =

(26 – 10)!

26!

16!26•25•24•23•22•21•20•19•18•17•16!

16!

13. How many 26-letter words can be made

if no repetition of a letter is allowed?

26!

14) How ways can your homeroom (of 23 people) choose an ASC rep and a ASC alternate?

23 P2 =

23 • 22 =

506

15) Suppose we just want to select 2 people in the homeroom to serve on the ASC committee. How many 2-person groups are possible

23 C2 =

23! =

21! 2!

23 • 22 =

2

253

16) How many 5-card “hands” are possible when dealt from a

deck of 52 cards?

52 C5 =

52! =

47! 5!52 • 51 • 50 • 49 • 48 • 47! =

47! • 5 • 4 • 3 • 2 • 1

2,598,960

17. Eight points are located on the circumference of a circle.

You want to draw a triangle whosevertices are each one of these points.How many triangles are possible?_______ • _______Starting CircleVertex___7!____ • ___6!____ 5040 • 7203,628,800

18) Out of a class of 6 seniors and 5 juniors. I need to select a dance committee that must contain 2 seniors and 1 junior. How many different ways can this be done?

6 C2 • 5 C1 =

6! • 5! =

4! 2! 4! 1!

6 • 5 •4! • 5 • 4! =

4! 2 4!

75