Introduction to Discrete
Mathematics
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Distinct Pirates - Discrete Math - Lecture Slides, Slides of Discrete Mathematics
Some concept of Discrete Math are Unique Path, Addition Rule, Clay Mathematics, Complexity Theory, Correspondence Principle, Discrete Mathematics, Group Theory, Random Variable, Major Concepts. Main points of this lecture are: Distinct Pirates, Rearrange, Places, Pretend, Permutations, Pretending, Arrangement, Rearrangements, Arrange, Carnegiemellon
Typology: Slides
2012/2013
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Introduction to Discrete
Mathematics
How many ways to rearrange
the letters in the word
“SYSTEMS”?
SYSTEMS
Let’s pretend that the S’s are distinct: S 1 YS 2 TEMS 3
There are 7! permutations of S 1 YS 2 TEMS 3
But when we stop pretending we see that we have counted each arrangement of SYSTEMS 3! times, once for each of 3! rearrangements of S 1 S 2 S 3
7! 3!
Arrange n symbols: r 1 of type 1, r 2 of
type 2, …, rk of type k
n r (^1)
n-r (^1) r (^2)
n - r 1 - r 2 - … - r (^) k- r (^) k
(n-r 1 )! (n-r 1 -r 2 )!r 2!
n! (n-r 1 )!r 1!
n! r 1 !r 2! … r (^) k!
Remember:
The number of ways to
arrange n symbols with r 1
of type 1, r 2 of type 2, …,
r k of type k is:
n!
r 1 !r 2! … r k!
5 distinct pirates want to divide 20 identical, indivisible bars of gold. How many different ways can they divide up the loot?
Sequences with 20 G’s and 4 /’s
How many different ways to divide
up the loot?
How many different ways can n
distinct pirates divide k identical,
indivisible bars of gold?
n + k - 1
n - 1
n + k - 1
k
How many integer solutions to
the following equations?
x 1 + x2 + x 3 + … + x n = k
x 1 , x 2 , x 3 , …, x n ≥ 0
n + k - 1
n - 1
n + k - 1
k
Identical/Distinct Dice
Suppose that we roll seven dice
How many different outcomes are
there, if order matters? 6
What if order doesn’t matter? (E.g., Yahtzee)
(Corresponds to 6 pirates and 7 bars of gold)
Counting Multisets
n + k - 1
n - 1
n + k - 1
k
There number of ways to choose a multiset of size k from n types of elements is:
( +^ + ) ( + ) =
+ + + + +
Polynomials Express
Choices and Outcomes
Products of Sum = Sums of Products
There is a correspondence
between paths in a choice
tree and the cross terms of
the product of polynomials!
1 X 1 X 1 X 1 X
1 X (^1) X
1 X
1 X X^ X^2 X^ X^2 X^2 X^3
Choice Tree for Terms of (1+X)^3
Combine like terms to get 1 + 3X + 3X^2 + X^3