Counting - H-SC
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Transcript Counting - H-SC
Counting Subsets of a
Set: Combinations
Lecture 28
Section 6.4
Wed, Mar 2, 2005
r-Combinations
An r-combination of a set of n elements is a
subset of r of the n elements.
The order of the elements does not matter.
The 3-combinations of the set {a, b, c, d, e}
are {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a,
c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e},
{c, d, e}.
Counting r-Combinations
Theorem: The number of r-combinations
of a set of n elements is
C(n, r) = n!/[r!(n – r)!].
Examples:
C(4, 2) = (4 3)/(2 1) = 6.
C(10, 3) = (10 9 8)/(3 2 1) = 120.
C(1000, 2) = (1000 999)/(2 1) = 499500.
Some Useful Facts
C(n, 0) = 1 for all n 0.
C(n, 1) = n for all n 1.
Notice that C(n, r) = C(n, n – r).
For example,
C(100, 99) = C(100, 1) = 100/1 = 100.
Therefore,
C(n, n) = 1 for all n 0.
C(n, n – 1) = n for all n 1.
Another Useful Fact
The TI-83 will calculate C(n, r).
Enter n.
Select MATH > PRB > nCr.
Enter r.
Press ENTER.
The value of C(n, r) appears.
Example: Counting rCombinations
In Math 121, I used to collect 48 daily
homework assignments.
Some assignments count more than
others.
I drop the “lowest” 4 homework grades.
Which should be dropped: 0 out of 30 or
15 out of 40?
I drop the 4 grades that hurt the student’s
average the most.
Example: Counting rCombinations
How can that be determined?
Can a computer program make the
determination by brute force (exhaustive
checking) within a reasonable amount of
time?
There are C(48, 4) = 194,580 possible
choices.
A computer can do the math really fast, in
say one second.
Example: Counting rCombinations
What if I dropped 6 grades?
There would be C(48, 6) = 12,271,512
possible choices.
Over 60 times as many.
This would require about 60 secs = 1 min.
Example: Counting rCombinations
What if I dropped 12 grades?
There would be C(48, 12) = over 69 billion
choices!
More than 350,000 as many!
This would require almost 350,000 sec =
over 4 days.
Example: Counting rCombinations
What if I dropped 44 grades!!!???
This must involve unimaginably many
possibilities!
How many would it be?
Example: Lotto South
In Lotto South (Loot the South?), a player
chooses 6 numbers from 1 to 49.
Then the state chooses at random 6
numbers from 1 to 49.
The player wins according to how many of
his numbers match the ones the state
chooses.
See the Lotto South web page.
Example: Virginia Lottery
There are C(49, 6) = 13,983,816 possible
choices.
Match all 6 numbers
There is only 1 winning combination.
Probability of winning is
1/13983816 = 0.00000007151
(it ain’t gonna happen)
Example: Virginia Lottery
Match 5 of 6 numbers
There are 6 winning numbers and 43 losing
numbers.
Player chooses 5 winning numbers and 1
losing numbers.
Number of ways is C(6, 5) C(43, 1) = 258.
Probability is 0.00001845 (forget about it).
Example: Virginia Lottery
Match 4 of 6 numbers
Player chooses 4 winning numbers and 2
losing numbers.
Number of ways is C(6, 4) C(43, 2) =
13545.
Probability is 0.0009686 (nope).
Example: Virginia Lottery
Match 3 of 6 numbers
Player chooses 3 winning numbers and 3
losing numbers.
Number of ways is C(6, 3) C(43, 3) =
246820.
Probability is 0.01765 (it just might
happen.)
Example: Virginia Lottery
Match 2 of 6 numbers
Player chooses 2 winning numbers and 4
losing numbers.
Number of ways is C(6, 2) C(43, 4) =
1851150.
Probability is 0.1324 (it will happen).
Example: Virginia Lottery
Match 1 of 6 numbers
Player chooses 1 winning numbers and 5
losing numbers.
Number of ways is C(6, 1) C(43, 5) =
3011652.
Probability is 0.4130 (it will happen).
Example: Virginia Lottery
Match 0 of 6 numbers
Player chooses 6 losing numbers.
Number of ways is C(43, 6) = 2760681.
Probability is 0.4360 (it will happen).
Example: Virginia Lottery
Note also that the sum of these integers is
13983816.
Note also that the lottery pays out a prize
only if the player matches 3 or more
numbers.
Match 3 – win $5.
Match 4 – win $75.
Match 5 – win $1000.
Match 6 – win millions.
Example: Virginia Lottery
Given that a lottery player wins a prize,
what is the probability that he won the $5
prize?
P(he won $5, given that he won)
= P(match 3)/P(match 3, 4, 5, or 6)
= 0.01765/0.01864
= 0.9469.
Example
Theorem (The Vandermonde convolution):
For all integers n 0 and for all integers r
with 0 r n,
r n r n
k 0 k r k
r
r
Proof: See p. 362, Sec. 6.6, Ex. 18.
Permutations of Sets with
Repeated Elements
Theorem: Suppose a set contains n1
indistinguishable elements of one type, n2
indistinguishable elements of another type,
and so on, through k types, where
n1 + n2 + … + nk = n.
Then the number of (distinguishable)
permutations of the n elements is
n!/(n1!n2!…nk!).
Proof of Theorem
Proof:
Rather than consider permutations per se,
consider the choices of where to put the
different types of element.
There are C(n, n1) choices of where to
place the elements of the first type.
Proof of Theorem
Proof:
Then there are C(n – n1, n2) choices of
where to place the elements of the second
type.
Then there are C(n – n1 – n2, n3) choices of
where to place the elements of the third
type.
And so on.
Proof, continued
Therefore, the total number of choices, and
hence permutations, is
C(n, n1) C(n – n1, n2) C(n – n1 – n2, n3)
… C(n – n1 – n2 – … – nk – 1, nk)
= …(some algebra)…
= n!/(n1!n2!…nk!).
Example
How many different numbers can be
formed by permuting the digits of the
number 444556?
6!/(3!2!1!) = 720/(6 2 1) = 60.
Example
How many permutations are there of the
letters in the word MISSISSIPPI?
11!/(4!4!2!1!) = 34650.