Combinatorics

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Combinatorics

• CSC 172
• SPRING 2002
• LECTURE 11

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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?

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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.

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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

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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

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• 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

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Permutations

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

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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.

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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

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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)!

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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

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In general

• n choose m

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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

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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

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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!)
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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)

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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
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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

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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

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In general

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

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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

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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

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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
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Several kinds of items into Bins

• Order in bin can be either important (queues) or
  not

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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
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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