CS 173: Discrete Mathematical Structures

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CS 173Discrete Mathematical Structures

• Cinda Heeren
• heeren_at_cs.uiuc.edu
• Siebel Center, rm 2213
• Office Hours M 11a-12p

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CS 173 Announcements

• Homework 8 available. Due 10/31, 8a.
• Midterm 2 11/9, 7-9p. Email me with conflicts.

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CS173 Permutations
Suppose you have time to listen to 10 songs on
your daily jog around campus. There are 6 Cake
tunes, 8 Moby tunes, and 3 Eagles tunes to choose
from. How many different jog playlists can you ma
ke?
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CS173 Permutations
Suppose you have time to listen to 10 songs on
your daily jog around campus. There are 6 Cake
tunes, 8 Moby tunes, and 3 Eagles tunes to choose
from. Now suppose you want to listen to 4 Cake, 4
Moby, and 2 Eagles tunes, in that band order.
How many playlists can you make?
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CS173 Permutations
Suppose you have time to listen to 10 songs on
your daily jog around campus. There are 6 Cake
tunes, 8 Moby tunes, and 3 Eagles tunes to choose
from. Finally, suppose you still want 4 Cake, 4 M
oby, and 2 Eagles tunes, and the order of the
groups doesnt matter, but you get dizzy and fall
down if all the songs by any one group arent
played together. How many playlists are there now
?
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CS173 Permutations
In how many ways can 5 distinct Martians and 3
distinct Jovians stand in line, if no two Jovians
stand together?
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CS173 Combinations
A combination is an unordered selection of
elements from some set.
The number of combinations of r distinct objects
chosen from n distinct objects is denoted by
C(n,r) or nCr or , and is read n choose r.
C(n,r) P(n,r)/r! n!/((n-r)!r!)
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CS173Combinations
A basketball squad consists of 12 players, 5 of
which make up a team. How many different teams
of players can you make from the 12?
Whats the diff?
In a running race of 12 sprinters, each of the
top 5 finishers receives a different medal. How
many ways are there to award the 5 medals?
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CS173 Combinations
A committee of 8 students is to be selected from
a class consisting of 19 frosh, and 34 soph.
In how many ways can 3 frosh and 5 soph be selec
ted?
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CS173 Combinations
A committee of 8 students is to be selected from
a class consisting of 19 frosh, and 34 soph.
In how many ways can a committee with exactly 1
frosh be selected?
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CS173 Combinations
A committee of 8 students is to be selected from
a class consisting of 19 frosh, and 34 soph.
In how many ways can a committee with at most 1
frosh be selected?
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CS173 Combinations
A committee of 8 students is to be selected from
a class consisting of 19 frosh, and 34 soph.
In how many ways can a committee with at least 1
frosh be selected?
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CS 173 Binomial Coefficients

• (a b)2 a2 2ab b2
• (a b)3 a3 3ab2 3a2b b3
• (a b)4 a4 4ab3 6a2b2 4a3b b4

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CS 173 Binomial Coefficients

• (a b)2 a2 2ab b2
• (a b)3 a3 3a2b 3ab2 b3
• (a b)4 a4 4a3b 6a2b2 4ab3 b4

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CS 173 Binomial Coefficients

• (a b)4 (a b)(a b)(a b)(a b)

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CS 173 Binomial Coefficients

• What is the coefficient of a8b9 in the expansion
  of (3a 2b)17?

What is n?
What is j?
What is x?
What is y?
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CS 173 Binomial Coefficients

• (a b)4 (a b)(a b)(a b)(a b)

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CS 173 Binomial Coefficients

• Sum each row of Pascals Triangle

Two proofs that
Suppose you have a set of size n. How many
subsets does it have?
How many subsets of size 0 does it have?
How many subsets of size 1 does it have?
How many subsets of size 2 does it have?
Count all subsets in this way, and we have the
result!
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CS 173 Binomial Coefficients

• Sum each row of Pascals Triangle

Two proofs that
Let x1 and y1 in Binomial Theorem.
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CS 173 Pascals Identity

• A relationship between the entries in Pascals ?.

Suppose T is a set, Tn. Let a be an element
in T, and let S T - a. Lets count the nCj
subsets of size j. Note that some of these
contain a, and some dont. How many contain a?
How many dont?
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CS 173 Vandermondes Identity

• Let m, n, and r be nonnegative integers with r
  not exceeding either m or n. Then

To choose r items, take some from A and some from
B. All possible ways of doing this gives the
result.
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CS 173 Combinations with repetition

• Suppose you want to buy 5 bags of chips from the
  3 kinds you like at Meijer. In how many
  different ways can you stock up?

Out of 7 items, we are choosing 2 to be bars.
From that, and our understanding of the model, we
can report the answer.

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CS 173 Combinations with repetition

• There are nr-1Cr, r-sized combinations from a
  set of n elements when repetition is allowed.

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CS 173 Permutations with indistinguishable
objects

• How many different strings can be made from the
  letters in the word rat?

How many different strings can be made from the
letters in the word egg?
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CS 173 Permutations with indistinguishable
objects
How many different strings can be made from the
letters in the phrase nano-nano?
Key thoughts 8 positions, 3 kinds of letters to
place.
In how many ways can we place the ns?
In how many ways can we place the as?
In how many ways can we place the os?
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CS 173 Permutations with indistinguishable
objects
How many distinct permutations are there of the
letters in the word APALACHICOLA?
How many if the two Ls must appear together?
How many if the first letter must be an A?
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CS 173 A little practice
A turtle begins at the upper left corner of an n
x m grid and meanders to the lower right corner.
How many routes could she take if she only moves
right and down?
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CS 173 A little practice
A turtle begins at the upper left corner of a m x
n grid and meanders to the lower right corner.
How many routes could she take if she only moves
right and down, and if she must pass through the
dot at point (a,b)?
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CS 173 A little practice
In how many ways can 11 identical computer
science books and 8 identical psychology books be
distributed among 5 students?
Hint forget about the psychology books for the
moment.
Hint how can you combine your soln for the CS
books with your soln for the Psych books?
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CS 173 A little practice
In an RNA chain of 20 bases, there are 4 As, 5
Us, 6 Gs, and 5Cs. If the chain begins either AC
or UG, how many such chains are there?
Let A denote the set of chains beginning with AC,
and U denote the set of chains beginning with UG.
Count them separately, and then sum.
First find A
18 bases, 3 As, 5 Us, 6 Gs, and 4Cs.
(This is like the MISSISSIPPI problem.)
A 18!/(3!5!6!4!)