CS6015: Linear Algebra and Random Processes

Lecture 25:  Counting collections

Learning Objectives

What are collections and how do you count them?

Collections

Recap

In a sequence order matters

cat       act

even though both words have the same letters: {t, c, a}

\neq

Collections

In a collection order does not matter

cat  = cta = act = atc = tac = tca

all the 6 words have the same letters: {t, c, a}

\frac{n!}{(n-k)!}

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

k = 3

How many sequences of 3 letters can you form (no repetition)?

How many collections of 3 letters can you form (no repetition)?

We don't know!

But we know how to count sequences! Can we reuse that knowledge?

Sequences: Breaking it down

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

26

3

Step 1: select the 3 letters to be put in the word

Step 2: re-arrange the 3 letters in 3! ways

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

k

Step 1: select the k items to be put in the sequence

Step 2: re-arrange the k items in k! ways

Sequences: Breaking it down

Making a collection

Re-arranging elements in the collection

N = number of ways of selecting k elements

k! = number of ways of re-arranging the k terms

Sequences: Breaking it down

Making a collection

Re-arranging elements in the collection

Number of sequences

= N * k!
= \frac{n!}{(n-k)!}
\therefore N = \frac{n!}{(n-k)!k!}

What is the number of ways of selecting 3 vowels from 5 vowels ?

Collections Sequences
(a,e,i) {(a,e,i), (a,i,e), (e,a,i), (e,i,a), (i,a,e),(i,e,a)}
(a,e,o) {(a,e,o), (a,o,e), (e,a,o), (e,o,a), (o,a,e), (o,e,a)}
(a,e,u) {(a,e,u), (a,u,e), (e,a,u), (e,u,a), (u,a,e), (u,e,a)}
(a,i,o) {(a,i,o), (a,o,i), (i,a,o), (i,o,a), (o,a,i), (o,i,a)}
(a,i,u) {(a,i,u), (a,u,i), (i,a,u), (i,u,a), (u,a,i), (u,i,a)}
(a,o,u) {(a,o,u), (a,u,o), (o,a,u), (o,u,a), (u,a,o), (u,o,a)}
(e,i,o) {(e,i,o), (e,o,i), (i,e,o), (i,o,e), (o,e,i), (o,i,e)}
(e,i,u) {(e,i,u), (e,u,i), (i,e,u), (i,u,e), (u,e,i), (u,i,e)}
(e,o,u) {(e,o,u), (e,u,o), (o,e,u), (o,u,e), (u,e,o), (u,o,e)}
(i,o,u) {(i,o,u), (i,u,o), (o,i,u), (o,u,i), (u,i,o), (u,o,i)}

a e i o u

10

60

\frac{5!}{~2!~3!~}

3!

Sequences
{(a,e,i), (a,i,e), (e,a,i), (e,i,a), (i,a,e),(i,e,a)}
{(a,e,o), (a,o,e), (e,a,o), (e,o,a), (o,a,e), (o,e,a)}
{(a,e,u), (a,u,e), (e,a,u), (e,u,a), (u,a,e), (u,e,a)}
{(a,i,o), (a,o,i), (i,a,o), (i,o,a), (o,a,i), (o,i,a)}
{(a,i,u), (a,u,i), (i,a,u), (i,u,a), (u,a,i), (u,i,a)}
{(a,o,u), (a,u,o), (o,a,u), (o,u,a), (u,a,o), (u,o,a)}
{(e,i,o), (e,o,i), (i,e,o), (i,o,e), (o,e,i), (o,i,e)}
{(e,i,u), (e,u,i), (i,e,u), (i,u,e), (u,e,i), (u,i,e)}
{(e,o,u), (e,u,o), (o,e,u), (o,u,e), (u,e,o), (u,o,e)}
{(i,o,u), (i,u,o), (o,i,u), (o,u,i), (u,i,o), (u,o,i)}

Sequence or Collection?

15

P

VP

T

S

ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA
CABD
CADB
CBAD
CBDA
CDAB
CDBA
DABC
DACB
DBAC
DBCA
DCAB
DCBA

Given a class of 15 students, in how many ways can you form a committee of 4 members?

All these 4! = 24 sequences are equal in a collection

15

All these 4! = 24 sequences are equal in a collection

ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA
CABD
CADB
CBAD
CBDA
CDAB
CDBA
DABC
DACB
DBAC
DBCA
DCAB
DCBA
\frac{15!}{~~~~~(15-4)!~~~~~~}
\frac{~}{4!}
n \choose k

The number of ways of selecting kkk objects from given nnn objects is                     and is denoted as           n(n−.       1)(n−2)...(n−k+1)n (n-1) (n-2) ... (n-k+1)

\frac{n!}{(n-k)!k!}

Collections

(some examples)

 

10

Consider 10 people in a meeting. If each person shakes hands with every other person in the room what is the total number of handshakes ?

Consider 10 people in a meeting. If each person shakes hands with every other person in the room what is the total number of handshakes ?

10

2

{10 \choose 2} = 45

You are going on a vacation and your suitcase has space for 3 shirts only? In how many ways can you fill the suitcase?

3

10

You are going on a vacation and your suitcase has space for 3 shirts only? In how many ways can you fill the suitcase?

3

10

You are going on a vacation and your suitcase has space for 3 shirts only? In how many ways can you fill the suitcase?

3

10

{10 \choose 3} = 120

There are 6 points on a 2 - dimensional plane such that no 3 points are collinear. How many segments can you draw from these 6 points?

6

2

{6 \choose 2} = 15

How many triangles can be formed from the vertices of a polygon of n sides?

n = 8

k = 3

{n \choose 3}

Collections

(with repetitions)

 

Recap

Sequences

without repetitions

with repetitions

Collections

without repetitions

with repetitions

\frac{n!}{(n-k)!}
n^k
\frac{n!}{(n-k)!k!}
?

How many breakfast combos containing 5 items can you form if you are allowed to have multiple servings of the same dish?

5

10

How many breakfast combos containing 5 items can you form if you are allowed to have multiple servings of the same dish?

5

How many breakfast combos containing 5 items can you form if you are allowed to have multiple servings of the same dish?

5

How many breakfast combos containing 5 items can you form if you are allowed to have multiple servings of the same dish?

5

How many breakfast combos containing 5 items can you form if you are allowed to have multiple servings of the same dish?

5

How many breakfast combos containing 5 items can you form if you are allowed to have multiple servings of the same dish?

5

{14 \choose 5}

n original counters

{n+k-1 \choose k}

k-1 magic counters

The number of ways of selecting kkk objects from given nnn objects with repetitions is             n(n−.       1)(n−2)...(n−k+1)n (n-1) (n-2) ... (n-k+1)

{n+k-1 \choose k}

(replicate any item which gets selected)

Summary

Sequences

without repetitions

with repetitions

Collections

without repetitions

with repetitions

\frac{n!}{(n-k)!}
n^k
\frac{n!}{(n-k)!k!}
{n+k-1 \choose k}

Collections

(+ multiplication principle)

 

Given a class of 7 boys and 8 girls, in how many ways can you form a committee of 4 members with 2 boys and 2 girls?

8

7

{8 \choose 2}
{7 \choose 2}
*

batsmen

keepers

pacers

spinners

7   

2   

4   

3   

Available

16

5  

1

3

2

Select

11

{7 \choose 5}
{2 \choose 1}
{4 \choose 3}
{3 \choose 2}
{7 \choose 5}
{2 \choose 1}
{4 \choose 3}
{3 \choose 2}
*
*
*

Total =   

(n−.       1)(n−2)...(n−k+1)n (n-1) (n-2) ... (n-k+1)

m_1 + m_2 + \dots + m_i = n
k_1 + k_2 + \dots + k_i = k
Given:n~items~of~i~different~types
{m_1 \choose k_1} * {m_2 \choose k_2} * \cdots * {m_i \choose k_i}
Form:collection~of~k~items

Collections

(+ subtraction principle)

 

3  cardiologists

4  diabetologists

2  neurologists

5  gynaecologists

7  general physicians

In how many ways can you form a 4-member committee containing at least one gynaecologist?

(total 21 doctors)

A = all possible committees of 4 members

B = all possible committees containing at least one gynaecologist

C = all possible committees containing no gynaecologist

count(A) = {21 \choose 4}
count(C) = {16 \choose 4}
count(B) = {21 \choose 4} - {16 \choose 4}

3  cardiologists

4  diabetologists

2  neurologists

5  gynaecologists

7  general physicians

(total 21 doctors)

Learning Objectives

What are collections and how do you count them?

Summary

 

Multiplication Principle

p*q*r*...

num. of choices at each step should be independent

address constraints first

Subtraction Principle

"at least one"

\frac{n!}{(n-k)!}
n^k
\frac{n!}{(n-k)!k!}
{n+k-1 \choose k}

Sequences

Collections

without repetitions

with repetitions

without repetitions

with repetitions

without repetitions

with repetitions

+ multiplication principle

+ multiplication/subtraction principle