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MASTERMIND. Did anyone play the game over the weekend? Any thoughts on strategy?. Combinatorics Review. CSC 172 SPRING 2004 LECTURE 12. 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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Mastermind
MASTERMIND

  • Did anyone play the game over the weekend?

  • Any thoughts on strategy?


Combinatorics review

Combinatorics Review

CSC 172

SPRING 2004

LECTURE 12


Assignments with replacement
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?

    • What would make a better password?

      • 4 digit “pins”

      • 3 letter initials


In general
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.


Inductive definition
Inductive Definition

Basis:

W(1,k) == k

Induction:

If we have n+1 items, we assign the first one in one of k ways and the remaining n in W(n,k) ways


Recurrence
Recurrence

W(1,k) = k

W(n+1,k) = k * W(n,k)

T(1) = k

T(n) = k * T(n-1)

Easy expansion

T(n) = kn


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


Example: Are there more 4-digit pins, or 3 letter initials? symbols or strings of length 3 built from 5 symbols?

Length 4, from 10

“values” = {0,1,2,3,4,5,6,7,8,9}

“items” are the 4 positions

N = 104 = 10000

Length 3, from 26

“values” = {a,b,c,…,x,y,z}

“items” are the 3 positions

N = 263 = 17576


Mastermind1
MASTERMIND symbols or strings of length 3 built from 5 symbols?

  • How many codes?

  • N colors

  • M positions

    Can you build an array of unknown

    Dimensionality?


Exercise aside
Exercise (aside) symbols or strings of length 3 built from 5 symbols?

Can you write

public int blackPegs(

String [] correct, String[] guess){

// return the number of correct colors

// in the correct positions

}


Exercise aside1
Exercise (aside) symbols or strings of length 3 built from 5 symbols?

  • Can you write

    public int whitePegs(

    String [] correct, String[] guess){

    // return the number of correct colors

    // in the incorrect positions

    }


Permutations
Permutations symbols or strings of length 3 built from 5 symbols?

Example: “SCRABBLE”

- start with 7 letters (tiles)

- how many different ways can you arrange them

- we don’t care about the word’s legality


Scrabble
Scrabble symbols or strings of length 3 built from 5 symbols?

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, 7*6 = 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 = 7*6*5*…*1 = 7! = 5040

In general, there are n! permutations of n items.


Ordered selections
Ordered Selections symbols or strings of length 3 built from 5 symbols?

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 7*6 = 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


In general1
In general symbols or strings of length 3 built from 5 symbols?

P(n,m), the number of ways to pick a sequence of m things out of n

== n*(n-1)*(n-2)*…*(n-m+1)

== n!/(n-m)!


Combinations
Combinations symbols or strings of length 3 built from 5 symbols?

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 7!/(7-4!) = 840

However, we don’t care about order.

So, how many ways are there to order 4 items?

4! = 24

840/24==35

== 7!/((7-4)!4!) = 35


In general2
In general symbols or strings of length 3 built from 5 symbols?

“n choose m”


Recursive definition for n choose m
Recursive Definition for n choose m symbols or strings of length 3 built from 5 symbols?

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


Inductive definition1
Inductive Definition symbols or strings of length 3 built from 5 symbols?

Basis:

for all n

there is only one way to choose all or none of the elements

Induction:

for 0 < m < n


Proving inductive definition direct definition
Proving symbols or strings of length 3 built from 5 symbols?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!)


Basis
Basis symbols or strings of length 3 built from 5 symbols?

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)


Induction
Induction symbols or strings of length 3 built from 5 symbols?

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


Reminders
Reminders symbols or strings of length 3 built from 5 symbols?

  • Midterm is Tuesday – in class

  • Project is due before you go on Spring break

  • For workshop read Weiss Section 13.1

    • The Josephus Problem

  • I will be out, most of next week

    • Thursday’s guest lecture will involve “game AI”

    • 7.7 & 10.2 (alpha-beta pruning)


Midterm
Midterm symbols or strings of length 3 built from 5 symbols?

  • Closed book, notes

  • Calculators will not be necessary – no laptops

  • Proof by induction

  • Solve recurrence relations

  • Big-Oh proof

  • Big-Oh analysis of some code

  • Linked list programming

  • Stacks, queues, arrays

  • Recursion/backtracking programming analysis

  • Sorting (merge, quick, insertion, shell)

  • Combinatorics


Orders with some equivalent items
Orders With Some Equivalent Items symbols or strings of length 3 built from 5 symbols?

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


How much sameness
How much “sameness” symbols or strings of length 3 built from 5 symbols?

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/(2*6) = 420


In general3
In general symbols or strings of length 3 built from 5 symbols?

The orders of n items with groups i1,i2,…,ik equivalent items is

n!/(i1!i2!..ik!)


Items into bins
Items into Bins symbols or strings of length 3 built from 5 symbols?

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


Trick
Trick symbols or strings of length 3 built from 5 symbols?

Imagine 5 markers, denoted “*” that represent separation between bins, and 7 tokens “T” that represent dice

The string “*TT**TTT*T*T” 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


In general4
In general symbols or strings of length 3 built from 5 symbols?

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 n+m-1

positions for the tokens


Several kinds of items into bins
Several kinds of items into Bins symbols or strings of length 3 built from 5 symbols?

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


Several kinds of items order within bin unimportant
Several kinds of items, order within bin unimportant symbols or strings of length 3 built from 5 symbols?

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


Order important
Order important symbols or strings of length 3 built from 5 symbols?

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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