Initial set = {issues}

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{H,T}

Object

Initial set = {issues}

Purpose : being able to count the outcomes of an experiment

e.g. : heads or tails

EXPERIMENT Sample space = {outcomes} : # ?

Sample space =

{HH, HT, TH, TT} : # = 4 Toss twice

e.g. : roll a die

{1,2,3,4,5,6}

Sample space =

{111, 112, 113, …} : # = ? Roll 3 times

1. Combinatorics

(2)

Reasoning : different cases ?

Order ? Y/N

Repetition ? Y/N n = # of available elements

p = # of elements to be chosen

Total number of possible outcomes

e.g. : heads or tails

{H,T} : n = 2

Example of outcome :

Toss 3 times p = 3

e.g. : deck of 52 playing cards

{…} : n = 52

Example of outcome :

take 3 cards p = 3

T T H

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1.2.1 p-lists

Order ? Yes

Repetition ? Yes n = # of elements of the initial set = 5

p = # of elements to be taken = 2

Outcomes are named « p-lists » (here : 2-lists) Initial set :

{1,2,3,4,5} : n = 5

Outcomes : Form 2-long numbers

p = 2

Number of possible outcomes : 25

1. Combinatorics

(4)

Let’s draw a CHOICE TREE :

1st figure

1 2 3 4 5

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

2nd figure

This 2-leveled tree shows its 25 ends :

5 times 5 branches = 5²

Why 5 ? n = 5 Why 2 levels ? p = 2

Number of p-lists : 5² = n^p = 25

1.2.1 p-lists

(5)

ORDER

Y N

REPETITION Y p-lists : n^p N

Definition : a p-list is an ordered list formed with p elements taken from a set, with possible repetition.

Result : the number of possible p-lists from a set of n elements is n^p. Exercise 8 :

* How many ways for placing 2 objects into 3 drawers ?

* How many numbers of 4 figures only contain the figures 1, 2, 3 ?

* How many words of 5 letters taken from {a ; b ; e ; m ; i ; r ; o} ?

1. Combinatorics

1.2.1 p-lists

(6)

Exercise 8 :

How many ways for placing 2 objects into 3 drawers ?

How many numbers of 4 figures only contain the figures 1, 2, 3 ?

How many words of 5 letters taken from {a ; b ; e ; m ; i ; r ; o} ?

1.2.1 p-lists

t1 t2 t3

n = 3

p = 2

Initial set outcomes

At1 Bt3 At3 Bt1 At1 Bt1

= 2-lists

Nb outcomes = 3² =9

1 2 3

n = 3

p = 4

2312 2321

= 4-lists

Nb outcomes = 3⁴ =81

a b e m i r o

n = 7

p = 5

bimor

= 5-lists

Nb outcomes = 7⁵ =16807 biomr

biror

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1.2.2 Permutations

Order ? Yes

Repetition ? No n = # of elements of the initial set = 5

Outcomes are named « permutations » Initial set :

{1,2,3,4,5} : n = 5

Outcomes : Form 2-long numbers

With different figures p = 2

1. Combinatorics

(8)

Let’s draw a CHOICE TREE : 1st figure

1 2 3 4 5

2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 5 1 2 3 4

2nd figure

This 2-leveled tree shows its 20 ends :

5 times 4 branches = 5×4

Why 5 ? n = 5

Why 4 ? p = 2 ⇒ 2 levels no repetition

⇒ -1 possibility each next level

number of permutations : 5×4 = 20

1.2.2 Permutations

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ORDER

Y N

REPETITION Y p-lists : n^p N Permut.

Definition : a permutation is an ordered list formed with p different elements taken from a set.

Result : the number of possible permutations of p elements taken from a set of n elements is

P_n^p

P =_n^p ^n!

(n-p)!

1. Combinatorics

1.2.2 Permutations

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enter n key : OPTN

screen item : PROB screen item : nPr enter p

key : EXE Casio

enter n

key : MATH

screen item : PRB screen item : nPr

or Arrangements enter p

key : ENTER TI

1.2.2 Permutations

Exercise 9 :

* How many pairs representative/assistant from a group of 25 students?

* How many ways can 3 blocks be piled, taking them among 10 blocks of different colors?

* How many words, with 5 different letters in {a, b, e, m, i, r, o}?

(11)

1. Combinatorics

1.2.2 Permutations

Exercise 9 :

How many pairs representative/assistant from a group of 25 students?

Ways to pile 3 blocks, taking them among 10 blocks of different colors?

How many words, with 5 different letters in {a, b, e, m, i, r, o}?

E1 E2 … E25

n = 25

p = 2

E4 E12 E12 E4 E4 E4

= permutations

Nb outcomes = A²₂₅ =600

Rouge Vert Bleu Jaune…

n = 10

p = 3

RVJ VRJ

= permutations Nb outcomes =

a b e m i r o

n = 7

p = 5

bimor

= permutations Nb outcomes = biomr

biror VRR

3

A10 =720

5

A7 =2520

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1.2.3 Combinations

Order ? No

Repetition ? No n = # of elements of the initial set = 5

Outcomes are named « combinations » Initial set :

{1,2,3,4,5} : n = 5

Outcomes : Take 2 different figures

p = 2

(13)

A choice tree won’t help us. Let’s compare combinations and permutations : Combinations : {1,2} ; {1,3} ; {1,4} ; {1,5} ; {2,3} ; {2,4} ; {2,5} ; {3,4} ; {3,5} ; {4,5}

# = 10

(1,2) (2,1)

permutations

of the combination {1,2}

# = 2

(1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5) (4,5) (3,1) (4,1) (5,1) (3,2) (4,2) (5,2) (4,3) (5,3) (5,4)

whole set of permutations

# = 20 Number of combinations : = 10

1. Combinatorics

1.2.3 Combinations

(14)

ORDER

Y N

REPETITION Y p-lists : n^p

N Permut. Combin.

Definition : a combination is a set (no order) formed with p different elements taken from a set.

Result : the number of possible combinations of p elements taken from a set of n elements is

P_n^p

C =_n^p ^n!

p!(n-p)!

C^p_n

1.2.3 Combinations

(15)

enter n key : OPTN

screen item : PROB screen item : nCr enter p

key : EXE Casio

enter n

key : MATH

screen item : PRB screen item : nCr

or Combinaisons enter p

key : ENTER TI

1. Combinatorics

1.2.3 Combinations

Exercise 10 :

* How many couples of representatives from a group of 25 students?

* How many different hands of 8 cards from a deck of 32 playing cards?

* How many draws of 6 different integers, taking them between 1 and 49?

(16)

1.2.3 Combinations

Exercise 10 :

How many couples of representatives from a group of 25 students?

How many different hands of 8 cards from a deck of 32 playing cards?

How many draws of 6 different integers, taking them between 1 and 49?

E1 E2 … E25

n = 25

p = 2

E4 E12 E12 E4 E4 E4

= combinations

Nb outcomes = C²₂₅ =300

Aco Rtr 8pi Vca Rpi …

n = 32

p = 8

Rtr8pi…

8piRtr…

= combinations Nb outcomes =

1 2 3 … 48 49

n = 49

p = 6

13-2-7-21-9-43

= combinations

Nb outcomes = 13-7-2-21-9-43

13-2-2-21-9-43 8pi8pi…

8

C32 =10 518 300

6

C49 =13 983 816

(17)

1.2.4 Combinations in a partition

Initial set :

{A,B,C,D,E,F,G,H,I,J,K,L,M,N, O,P,Q,R,S,T,U,V,W,X,Y,Z} :

n = 26, n₁ = 20, n₂ = 6

Ω = {outcomes}

Draw three letters, simultaneously

p = 3

1. Combinatorics

A = {3C}

CRS TGV BTS PLS

…

B = {2C1V}

EDF DUT BAC PLI

… C = {1C2V}

FOU IUT ACE ELU

…

D = {3V}

EAU AIE YEA OUI

…

( )

^A ^C³²⁰ ^C⁰⁶ ^{1 140}

Card = × =

( )

^C³²⁶ ^{2 600}

Card Ω = =

( )

^B ^C²²⁰ ^C¹⁶ ^{1 140}

Card = × =

( )

^C ^C¹²⁰ ^C²⁶ ³⁰⁰

Card = × =

( )

^D ^C²⁰⁰ ^C³⁶ ²⁰

Card = × =