Announcements
Finite Probability
Wednesday, September 21st
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MyMathLab 3 is due Monday Sept 26
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Problem Set 3 is due Monday Sept 26
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Midterm 1 Extra Practice on MyMathLab
Today: Sec. 5.6: More Counting Methods II
Recognize useful ways to break down counting problems
Use the multiplication principle alongside
combinations/permutations to solve counting problems
Next Class: Sec. 5.7: Binomial Theorem
Cherveny
Sept 21
Math 1004: Probability
Identical Arrangements
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Recall: C (n, r ) = P(n,r
r ! . There were r ! ways to rearrange the r
objects chosen, so P(n, r ) over counted C (n, r ) by r ! times.
Example
In how many arrangements can 6 people sit around a round table?
Answer: 6!
6 = 5! = 120.
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There are 6! ways to arrange them against a wall. Seating
that lineup around a table forgets which of the six people was
first, so 6! over counts table arrangements 6-fold times.
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Alternatively, seat one person at the table (anywhere.. all
seats are the same). Now there are 5! ways to arrange
everybody with respect to that person.
Cherveny
Sept 21
Math 1004: Probability
Identical Arrangements
Example
How many different words can be made of the letters in BOSTON?
Answer: 6!
2! = 360 words. There are 6! ways to arrange 6 different
letters, and 2! ways to rearrange letters the two letters that are
identical.
Example
How many different words can be made of the letters in
MISSISSIPPI?
Answer:
Cherveny
11!
4!4!2!
= 34650 words
Sept 21
Math 1004: Probability
Poker Hands
Suppose you draw a 5 card hand from a standard 52 card deck.
(a) How many ways can you draw the 5 card hand?
(b) How many hands have exactly 3 spades?
(c) How many hands have exactly 2 kings?
(d) How many hands have at least one king?
(e) (Tricky) How many hands have exactly 2 kings or exactly 2
aces?
(f) How many hands are a flush (all the same suit)?
(g) How many hands have cards all of different ranks?
(h) How many hands are a full house (two of one rank and three
of another)?
(i) (Tricky) How many hands are a two pair (exactly two cards of
one rank, two of another rank, and one of a third rank)?
Cherveny
Sept 21
Math 1004: Probability
Poker Hands Answers
Answers:
(a) C (52, 5)
(b) C (13, 3) · C (39, 2)
(c) C (4, 2) · C (48, 3)
(d) C (52, 5) − C (48, 5)
(e) C (4, 2) · C (48, 3) + C (4, 2) · C (48, 3) − C (4, 2) · C (4, 2) · 44
(f) 4 · C (13, 5)
(g) C (13, 5) · 45
(h) 13 · C (4, 2) · 12 · C (4, 3)
[same as P(13, 2) · C (4, 2) · C (4, 3)]
(i) C (13, 2) · C (4, 2) · C (4, 2) · 11 · 4
Cherveny
Sept 21
Math 1004: Probability
Extra Challenge: Knights of King Arthur’s Court
10 knights in King Arthur’s court need to sit at a round table.
How many seating arrangements are there if...
(a) There are no seating restrictions?
(b) Sirs C, D, and E must all be seated together?
(c) Sir A and Sir B are feuding and cannot be seated next to each
other?
After the knights are seated, the dinners are brought out. There
are 3 venison, 4 duck, 2 boar, and a quail.
(d) How many ways can people get dinners? [Hint: BOSTON]
(e) How many ways can dinners be received if all the same types
of dinners are placed next to each other?
(f) How many ways can the dinners be assigned if Sir F can’t eat
duck?
Cherveny
Sept 21
Math 1004: Probability
Extra Challenge: King Arthur Answers
(a)
(b)
(c)
(d)
10!
10 = 9!
8!
8 · 3! = 7! · 3!
9! − 9!
9 · 2 (total
seating arrangements minus the ways A, B
can be seated together)
10!
3!4!2!
4!
4 · 10
(e)
(place the four types of dinner in order around a table
and then there are ten ways to rotate that arrangement)
(f)
10!
3!4!2!
Cherveny
9!
− 3!3!2!
(total number of ways to assign dinners minus
the ways dinners can be assigned if Sir F has been given a
duck)
Sept 21
Math 1004: Probability