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Combinatorics

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Title: Combinatorics


1
Combinatorics
  • CSC 172
  • SPRING 2002
  • LECTURE 11

2
Assignments With Replacement
  • Example Passwords
  • Are there more strings of length 5 built from
    three symbols or strings of length 3 built from 5
    symbols?

3
In general
  • We are given n items, to each we must assign
    one of k values
  • Each value may be used any number of times
  • Let W(n,k) be the number of ways
  • How many different ways may we assign values to
    the items?
  • Different ways means that one or more of the
    items get different values.

4
Inductive Definition
  • Basis
  • W(1,k) k
  • Induction
  • If we have n1 items, we assign the first one in
    one of k ways and the remaining n in W(n,k) ways

5
Recurrence
  • W(1,k) k
  • W(n1,k) k W(n,k)
  • T(1) k
  • T(n) k T(n-1)
  • Easy expansion
  • T(n) kn

6
  • Example Are there more strings of length 5 built
    from three symbols or strings of length 3 built
    from 5 symbols?
  • Length 5, from 3
  • values 0,1,2
  • items are the 5 positions
  • N 35 243
  • Length 3, from 5
  • values 0,1,2,3,4
  • items are the 3 positions
  • N 53 125

7
Permutations
  • Example SCRABBLE
  • - start with 7 letters (tiles)
  • - how many different ways can you arrange them
  • - we dont care about the words legality

8
Scrabble
  • We can pick the first letter to be any of the 7
    tiles
  • For each possible 1st letter, there are 6 choices
    of second letters
  • Or, 76 42 possible two letter prefixes
  • Similarly, for each of the 42, there are 5
    choices of the third letter. 42 5 210, and
    so on
  • Total choices 7651 7! 5040
  • In general, there are n! permutations of n items.

9
Ordered Selections
  • Suppose we want to begin Scrabble with a 4 letter
    word? How many ways might we form the word from
    our 7 distinct tiles?
  • For each possible 1st letter, there are 6 choices
    of second letters 76 42 possible two letter
    prefixes
  • For each of the 42, there are 5 choices of the
    third letter. 42 5 210
  • For each of the 120, there are 4 choices of the
    third letter. 210 4 840

10
In general
  • P(n,m), the nmber of ways to pick a sequence of m
    things out of n
  • n(n-1)(n-2)(n-m1)
  • n!/(n-m)!

11
Combinations
  • Suppose we give up trying to make a word and want
    to throw 4 of our 7 tiles back in the pile? How
    many different ways can we get rid of 4 tiles?
  • Ordered selection 4!/(7-4!) 840
  • However, we dont care about order.
  • So, how many ways are there to order 4 items?
  • 4! 24
  • 840/2435
  • 7!/((7-4)!4!) 35

12
In general
  • n choose m

13
Recursive Definition for n choose m
  • We want to choose m things out of n, we can
    either take or reject the first item.
  • If we take the first, then we can take the rest
    by choosing m-1 of the remaining n-1
  • We can do this in (n-1) choose (m-1) ways
  • OTOH, if we reject the first item, then we can
    get the rest by choosing m of the remaining n-1
  • We can do this in (n-1) choose m ways

14
Inductive Definition
  • Basis
  • for all n
  • there is only one way to choose all or none
    of the elements
  • Induction
  • for 0 lt m lt n

15
Proving Inductive Definition Direct Definition
  • What is the induction parameter?
  • Zero for the basis case
  • decreases in the inductive step
  • Complete induction on m(n-m)

Prove c(n,m) n!/((n-m)!m!)
16
Basis
  • If m(n-m) 0, then either m 0 or m n
  • If m 0,
  • then n!/(n-m)!m! n!/n! 1 c(n,0)
  • If m n,
  • then n!/(n-m)!m! n!/n! 1 c(n,n)

17
Induction
  • By definition c(n,m) c(n-1,m-1) c(n-1,m)
  • Assume,
  • c(n-1,m-1) (n-1)!/((n-m)!(m-1)!)
  • c(n-1,m) (n-1)!/((n-m-1)!m!)
  • Add the left sides c(n,m), by definition
  • Add the right sides n!/(n-m)!m!

, clearly
18
(No Transcript)
19
Orders With Some Equivalent Items
  • In real life, we play Scrabble with duplicate
    letters
  • Suppose you draw S,T,A,A,E,E,E at star, how
    many 7-letter words can you make.
  • Similar to permutations, but now, some are
    indistinguishable, because of duplicates
  • Trick we can mark the letters to make them
    distinguishable S,T,A1,A2,E1,E2,E3
  • Then we get 7!5040 ways

20
How much sameness
  • But some order are the same
  • E3TA1E1SA2E2 E3TA2E1SA1E2
  • The two As can be ordered in 2! 2 ways
  • The three Es can be ordered in 3! 6 ways
  • So, the number of different words is
  • 7!/2!3! 540/(26) 420

21
In general
  • The orders of n items with groups i1,i2,,ik
    equivalent items is
  • n!/(i1!i2!..ik!)

22
Items into Bins
  • Suppose we throw 7 dice (6 sided). How many
    outcomes are there?
  • Place each of 7 items into one of 6 bins
  • The tokes are the dice
  • The bins are then number of dice
  • Putting the second token into bin 3 means that
    the second die shows 3

23
Trick
  • Imagine 5 markers, denoted that represent
    separation between bins, and 7 tokens T that
    represent dice
  • The string TTTTTTT corresponds to
  • no 1s, two 2s, no 3s, three 4s, a 5 and a 6
  • How many such strings?
  • orders with identical items
  • 12 items, 5 of type , 7 of type T
  • 12!/5!7! 792

24
In general
  • The number of ways to assign n items to m bins
  • The number of orders of n-1 markers and n tokens

We are picking n out of the possible
nm-1 positions for the tokens
25
Several kinds of items into Bins
  • Order in bin can be either important (queues) or
    not

26
Several kinds of items, order within bin
unimportant
  • If we have items of k colors, with ij items of
    the jth color, then the number of distinguishable
    assignments into m bins is

Example 4 red dice, 3 blue dice, 2 green dice 6
bins
27
Order important
  • If there are m bins into which ij items are
    placed and order within the bin matters
  • Imagine m-1 markers separating the bins and ij
    tokens of the jth type
  • By orders with identical items
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