Counting and Probability: Factorials, Permutations, and Combinations, Slides of Engineering Mathematics

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Counting & Probability

BASIC COUNTING FUNCTIONS

• The 24 permutations of 1,2,3,4 in

lexicographical order (a.k.a the “dictionary

order”) :

Falling factorial

• 5 people enter a queue in a random manner, and the queue

size is only 3. How many different patterns can we observe?

• There are five combinations for the first position.
• After the first person has arrived, there are four combinations

for the second position.

• After the first and second persons have arrived, there are three

combinations for the third position.

• Answer = (5)(4)(3) = 60.

• Notation: (n)k = (n)(n-1)(n-2)…(n-k+1).

• (n)k is called the “falling factorial”, the “Pochhammer

symbol”, or “descending factorial”.

• When k=n, (n) n reduces to n!.

http://en.wikipedia.org/wiki/Pochhammer_symbol

Counting when order is irrelevant

• In how many ways can we choose 2 ice-cream

toppings out of 10 ice-cream toppings?

• Let’s call the 10 toppings A to J.

• There are 45 choices:

AB, AC, AD, AE, AF, AG, AH, AI, AJ,

BC, BD, BE, BF, BG, BH, BI, BJ, CD,

CE, CF, CG, CH, CI, CJ, DE, DF, DG,

DH, DI, DJ, EF, EG, EH, EI, EJ, FG,

FH, FI, FJ,GH, GI, GJ, HI, HJ, IJ.

Binomial coefficient

• We use the symbol to stand for the

number of ways we can choose k objects out

of n distinguishable objects.

• Example:

• It is pronounced as “n choose k”.

• Other notations : nCk, Cnk

http://en.wikipedia.org/wiki/Binomial_coefficient

Model of Balls and Bins

• Throw k balls into n bins. The n bins are

distinguishable

– The balls may be distinguishable or indistinguishable.

– There may be at most one ball per bin, or no such

restriction.

• Task: Count the total number of combinations in

each case

May be more than one ball per bin

• Throw k balls into n bins. The n bins are

distinguishable

– Balls are distinguishable

– Balls are indistinguisable

Throwing k Balls into n Bins

Balls Bins At most one ball per bin

May be more than one ball per bin

Distinguishable Distinguishable

Indistinguishable Distinguishable

Occupancy problem

• The number on the lower right corner of the

table is the solution to the so called

occupancy problem.

• Example: n=3, k = 2. There are 6 combinations

http://en.wikipedia.org/wiki/Occupancy_theorem

The questions on the board

• Paint the faces of a cube by six different

colours, so that each face has one and

only one colour. How many different

ways can we paint the cube?

• There are eight persons and six

different types of drinks. Each person

want to buy one drink. In how many

ways can they purchase?

• There are five beads of different

colours. In how many ways can we put

them together to form a bracelet?

A

B

C

Terminologies

• The end result of a random experiment is called an

outcome.

• A random experiment may be throwing a pair of dice, or

checking whether it is sunny or raining today.

• The set of all possible outcomes is called the sample

space of the random experiment.

• Each element in the sample space is an outcome.
• We usually use the Greek letter  for the sample space.

• An event is a set of outcomes.

• A probability measure assigns a real number to each

event.

• The probability of an event E is denoted by Pr( E ).

Venn diagram

Sample space

Event E 1

E 1 = { 1 , 3 }

Event E 2

E 2 = { 2 ,  3 ,  5 }

E 1 c= { 2 ,  4 ,  5 } (^) E 2 c^ = { 1 ,  4 }

Superscript c^ stands for set complement.