counting unit review sheet. 1. there are five choices of ice cream and three choices of cookies....
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![Page 1: 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](https://reader031.vdocuments.us/reader031/viewer/2022032313/56649e4c5503460f94b4213b/html5/thumbnails/1.jpg)
Counting Unit
Review Sheet
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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
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2. How many different 3-letter “words” can be formed from the
letters in the word CANOE? • • ____
5 • 4 • 3___
60
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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
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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
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5. How many different arrangements are there of the digits 166555? 6! =
2! 3!
6 • 5 • 4 • 3!
2 • 3!
60
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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
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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
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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
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13. Find the number of arrangements of the word LEVELED
7! =
3! 2!
7 • 6 • 5 • 4 • 3!
3! 2
840
2
420
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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
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(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
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(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
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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
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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
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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
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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
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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
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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
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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
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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
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9. How many ways can 7 people be seated in a circle?
(7-1)! =
720
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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
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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
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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!
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13. How many 26-letter words can be made
if no repetition of a letter is allowed?
26!
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14) How ways can your homeroom (of 23 people) choose an ASC rep and a ASC alternate?
23 P2 =
23 • 22 =
506
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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
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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
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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
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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