MULTIPLICATION PRINCIPLE

Determine all the possibility of the outcomes.If there’s x ways to do f1 task & y ways to do 2nd task..Then… there are ( xy) ways to do the procedure.

For example, if ice cream sundaes come in 5 flavors with 3 possible toppings, how many different sundaes can be made with one flavor of ice cream and one topping?

Solution : Instead of list all the possibilities,

5 • 3 = 15 possible sundaes

ADDITION PRINCIPLE

Let A and B be disjoint events that have different

outcomes. Then.. The total number of outcomes for

the event A and B is A + B.

Suppose that we want to buy a flower from two

flower shops. The first shop has 7 types of flowers

and the second has 9 types of flowers.

How many types of flowers that altogether we can

choose from?

Solution : 7 + 9 = 16

COMBINING OPERATIONS

Combining both the add and multiplication principle.

Suppose there are 7 types of roses and 4 types of

tulips in a gardens. There are also 4 types of

conifer in the garden. The neighbor’s gardener want

to build another garden that have flowering plants

and 4 types of conifer in another garden. How much

is the possible ways for the flowers in the garden to

be form?

Solution : (7+4) • 4 = 44 ways.

Definition:

Permutation:

An arrangement is called a Permutation. It is the rearrangement of

objects or symbols into distinguishable sequences. When we set things in

order, we say we have made an arrangement. When we change the order, we

say we have changed the arrangement. So each of the arrangement that can

be made by taking some or all of a number of things is known as

Permutation.

Combination:

A Combination is a selection of some or all of a number of different

objects. It is an un-ordered collection of unique sizes. In a permutation the

order of occurrence of the objects or the arrangement is important but in

combination the order of occurrence of the objects is not important.

PERMUTATION AND COMBINATION

Formula:

Permutation = nPr = n! / (n-r)!

Combination = nCr = nPr / r!

where,

n, r are non negative integers and r<=n.

r is the size of each permutation.

n is the size of the set from which elements are

permuted.

! is the factorial operator.

Example:

Find the number of permutations and combinations: n=6; r=4.

Step 1:

Find the factorial of 6.

6! = 6*5*4*3*2*1 = 720

Step 2:

Find the factorial of 6-4.

(6-4)! = 2! = 2

Step 3:

Divide 720 by 2.

Permutation = 720/2 = 360

Step 4: Find the factorial of 4.

4! = 4*3*2*1 = 24

Step 5:Divide 360 by 24.

Combination = 360/24 = 15

PIGEONHOLE PRINCIPLE

The Pigeonhole Definition : If k is a positive integer and k+1 or more objects

are placed into k boxes,then there is at least one box containing two or more of

the objects.

No of pigeon must greater than no of boxes/pigeonhole

Formula : 1) k([N/k]-1) < k( ([N/k]+1)-1 ) = N

k = no.of boxes/pigeonhole

N = no.of pigeon

: 2) N = k(r − 1) + 1

r = objects

Example : Among 100 people there are at least [100/12]=9 who were born in the

same month

- 4 ( require boxes) × 9 = 36

- 8 ( extra boxes) × 8 = 64

100