The document discusses several mathematical principles: 1) The multiplication principle states that if there are x ways to do one task and y ways to do another, there are xy total ways to complete both tasks. 2) The addition principle states that if events A and B have no shared outcomes, the total number of outcomes is the sum of A and B's individual outcomes. 3) The pigeonhole principle states that if k objects are put into k boxes, then at least one box will contain multiple objects.

1. 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

2. 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

3. 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.

4. 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

5. 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.

6. 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

7. 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