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Combinations
• In some problems (e.g. dealing cards) we do not care
about the order that the objects are in. In this case, we
deal with combinations rather than permutations:
– If order matters Permutations
– If order does not matter Combinations
• Since order doesn’t matter, then we are counting some
objects multiple times:
Here we take into account the fact that we are counting r
multiple times.
)!(!
!
rnr
n
r
nCrn
↑
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Does order really matter?
Say we have three balls in a box – one red, one black, and
one white; and we randomly choose 2 balls from the box,
one at a time:
If order matters, then totally we have 6 possible outcomes:
• {white, red}, {red, white}, {white, black}, {black,
white},
{red, black}, and {black, red }
If order does not matter, then totally we have 3
possible results:
• {white, red}, {white, black} and {black, red}
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Calculating Combinations
• There should be a button on most calculators to
perform permutations.
• There should also be a button allowing you to
perform combinations.
• If there is not, all calculators should have a factorial
button (!) which allows you to compute using the formula.
rnC
rnP
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Combinations example (Ia)
• A child has 7 different toys in his toy box. He is only
allowed to take three of his toys with him on a family
outing. How many different sets of toys can he take?
– Suppose his toys are T1, T2,T3, T4, T5, T6, T7
•Remember: order does NOT matter
– Therefore, choosing (T1& T2) is the same
as choosing (T2& T1)
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Combinations example (Ib)
Since there are few enough possibilities, let’s list them just for fun:
T1, T2 T2, T3 T3, T4 T4, T5 T5, T6 T6, T7T1, T3 T2, T4 T3, T5 T4, T6 T5, T7T1, T4 T2, T5 T3, T6 T4, T7T1, T5 T2, T6 T4, T7T1, T6 T2, T7T1, T7
Notice, for example, that T1,T2 is counted, but T2,T1 is not counted. This is because the order does not matter; therefore, they are treated as the same event.
1012
120
!3!2
!5
)!25(!2
!525
C
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Combinations example (II)
The U.S. Senate consists of 100 senators, 2 from each of the 50 states. A committee consisting of 5 senators is to be formed.
- How many different committees are possible?
- How many are possible if no state can have more than 1 senator on the committee?
520,287,755100 C
320,800,67!5
92*94*96*98*100
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Combinations example (III)
What is the probability to have at least one king in a 5-card poker hand from 52 cards?
- Number of poker hands?
- Number of hands with at least one king?[Hint: First find the number of hands with no king and then
find the compliment…]
960,598,2)!552(!5
!52552
C
304,712,1)!548(!5
!48548
C
656,886304,712,1960,598,2
341.0960,598,2
656,886P (at least one
king)9
Ordered Partitions
!!...!
!
,...,, 2121 kk mmm
m
mmm
m
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Ordered Partitions example (17)
Suppose you have six different types of flowers and
three planters. In how many ways can you put three
flowers in the first planer, two flowers in the second
planter, and one flower in the final planter?
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Ordered Partitions example (17b)
• Flowers: F1, F2, F3, F4, F5, F6• Planters: P1, P2, P3
Planter 1 Planter 2 Planter 3F1, F2, F3 F4, F5 F6F1, F2, F3 F4, F6 F5F1, F2, F3 F5, F6 F4F1, F2, F4 F3, F5 F6Etc….
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Ordered Partitions example (17c)
How many ways are there?
We can refer to this as a multinomial coefficient
60!1!2!3
!6
1,2,3
6
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Ordered Partitions example (18a)
What is the probability that in a game of Hearts, one
player gets all of the Hearts? Hearts is played with 4
people; each player is dealt 13 cards from a
standard 52-card deck.
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Ordered Partitions example (18b)
Total possible dealings:
One player gets all the hearts:
P(one player gets all hearts) =
!13!13!13!13
!52
13,13,13,13
52
!13!13!13
!39
13,13,13
39
1210*57.1!52
!39!13
!13!13!13!13!52!13!13!13
!39
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Remember…
• Always more than one way to solve a problem• If you don’t know which method to use to solve a problem, that’s OK! There is not always one definitive method!• A good approach is to imagine yourself doing the event described• When you want to count the possible outcomes, keep very close track to ALL the decisions you have to make and apply the basic counting rule
General rule of thumb: - choosing with replacement BCR - choosing without replacement, order matters
Permutations - choosing without replacement, order does not matter
Combinations
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More examples…
A fair coin is flipped 8 times, what is the probability to see exactly 4 heads in the 8 tosses?
First, find number of total possible flips:2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 28 = 256
Then, find possibilities with exactly four heads:NOT 2 * 2 * 2 * 2! Why not?
P(exactly 4 heads) = 70/256 = 0.273
8 4
8! 8!70
4!( )! 4!(4 4)!n k
nC C
k n k
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More examples…
What is the probability to be dealt a poker hand with two Aces?
First, find total number of poker hands:
Then, find possibilities with exactly 2 Aces:
P(2 aces) = 103,776/2,598,960 = 0.04
776,1033
48
2
4
52 52! 52!2,598,960
5 5!(52 5)! 5!47!
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More examples…
If five fair die are tossed, what is the probability they will all show different faces?
First, find total number of outcomes:6 * 6 * 6 * 6 * 6 = 65 = 7,776
Then, find possibilities (number of die) with different faces:
6 * 5 * 4 * 3 * 2 = 6! = 720
P(different faces) = 720/7,776 = 0.0926
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More examples…
4 tennis players (two teams of two) are to be chosen from
12 players at a club. Two of them really dislike each other
and refuse to be on the same court as the other one. How
many possible combinations of four players are possible?
[Hint: When and why do we add and/or multiply?]
We can add together the combinations of when those two
players aren’t on the court and when they are on the
court:
4502402101
2
3
10
4
10
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More examples…
What is the probability of winning a jackpot in
a lottery that requires that you have 5 correct
numbers out of 30 numbers?
000007.01
)(530
C
winP
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A final example…
An ordinary deck of 52 cards is shuffled and dealt. What is
the probability that:
(a) the seventh card is an ace?
(b) the first ace occurs on the seventh card dealt?
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A final example…
(a)the seventh card is an ace?
First, look at all 7 card combos:
[notice we used a permutation because order matters]
Then, the probability the last card is any specific card:
So we take:
P(7th card is an Ace) = 0769.013
1
!45
!52752 P
51 6
51!( )*4 ( )*4
(51 6)!P
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51!*4 145!
52! 1345!
A final example…
(b) the first ace occurs on the seventh card dealt?
Possible outcomes where 1st Ace is on the 7th card?
P(1st Ace is on 7th card) = = 0.0524
4*!42
!484*648 P
24
48!*4
42!52!45!
