counting unit review sheet. 1. there are five choices of ice cream and three choices of cookies....
TRANSCRIPT
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