This document discusses counting principles such as the product rule, sum rule, and subtraction rule. It also covers permutations and combinations. The product rule states that if a procedure can be broken into stages with m possible outcomes for the first stage and n for the second, the total number of ways to complete the procedure is m * n. Permutations refer to ordered arrangements and are calculated with n!/(n-r)!. Combinations refer to unordered arrangements and are calculated with n!/r!(n-r)!.

1. Permutation and
combination
Prepared by: Angela Clarito

2. Counting Principles
● PRODUCT RULE
➔ a procedure can be broken down into first and second
stages
➔ there are m possible outcomes for the first stage for
each of these outcomes,
➔ there are n possible outcomes for the second stage
➔ the total procedure can be carried out in m · n ways

3. Counting Principles
● PRODUCT RULE EXAMPLE
➔ If i have 4 different tshirt and 3
different trouser. How many different
outfits do i have?
➔ 4 * 3 = 12

4. Tree Diagram
A Visual strategy we can use to represent counting problems.
➔ If i have 4 different tshirt and 3
different trouser. How many different
outfits do i have?

5. Tree Diagram
➔ If i have 4 different tshirt and 3 different
trouser. How many different outfits do i have?
ts1
ts2
ts3
ts4
t1
t2
t3
ts1 t1
ts1 t2
ts1 t3
ts1 t4
.
.
.
12 Outfits
t1
t2
t3
t1
t2
t3
t1
t2
t3

6. Counting Principles
● SUM RULE
➔ a first task can be performed in m (distinct) ways
➔ a second task can be performed in n (distinct) ways
➔ the two tasks cannot be performed simultaneously
➔ performing either task can be accomplished in any one
of m + n ways

7. Counting Principles
● SUM RULE EXAMPLE
➔ If i want to take a trip. I can travel to
one of 37 international places or one of
the 12 domestic places. How many trip
vacation choices do i have?
➔ 37 + 14 = 51

8. Counting Principles
● SUBTRACTION RULE
➔ Suppose event E can occur n ways, event F can occur m
ways, and
➔ There are p ways that E and F both occur.
➔ Then there are n+m-p ways E or F can occur.
➔ The subtraction rule is also called inclusion-
exclusion principle.

9. Counting Principles
● SUBTRACTION RULE EXAMPLE

10. Counting Principles
● SUBTRACTION RULE EXAMPLE
➔ If a card is drawn from a standard 52-card
deck, how many cards are Kings or Hearts?
KINGS HEARTS
# of Kings + # of Hearts - # of (kings
and hearts)
4 + 13 - 1 = 16 that are kings or are
hearts

11. Permutation
● An ordered arrangement of
distinct objects. An r-
permutation is the
arrangement of r-elements
of a set .
● Mathematical technique that
determines the number of
possible arrangements in a
set when the order of the
arrangements matters.
ORDER IS IMPORTANT

12. PERMUTATION Formula
P(n,r) = n!
(n-r)!
Denotes as permutation of n objects taken r
at a time .
● PERMUTATION EXAMPLE
➔ 5 students are to be chosen and seated in a
3 chairs. How many ways are possible.

13. PERMUTATION Formula
P(n,r) = n!
(n-r)!
● PERMUTATION EXAMPLE
➔ 5 students are to be chosen and seated in a
3 chairs. How many ways are possible.
P (5, 3) = n!
(n-r)!
P (5, 3) = 5!
(5-3)!
P (5, 3) = 5!
2!
P (5, 3) = 5.4.3.2.1
2.1
Simplify
P (5, 3) = 5.4.3.2!
2!
Simplify
P (5, 3) = 60 possible ways

14. PERMUTATION Formula
P(n,r) = n!
(n-r)!
● PERMUTATION EXAMPLE
➔ Suppose we have 6 different stuffed toys
and we wish to arrange 4 of them in a row.
How many ways can this be done?
P (6, 4) = n!
(n-r)!
P (6, 4) = 6!
(6-4)!
P (6, 4) = 6!
2!
P (6, 4) = 6.5.4.3.2.1
2.1
Simplify
P (6, 4) = 6.5.4.3.2!
2!
Simplify
P (6, 4) = 360 possible ways

15. COMBINATION
● An unordered arrangement of
set. An r-combination is a
subset of the set with r
elements.
● Mathematical technique that
determines the number of
possible arrangements in a
collection of items where
the order of selection does
not matter.
ORDER IS NOT IMPORTANT

16. COMBINATION Formula
C(n,r) = n!
r!(n-r)!
Denotes as combination of n objects taken
r at a time .
● PERMUTATION EXAMPLE
➔ Evaluate C(4, 3)

17. COMBINATION Formula
C(n,r) = n!
r!(n-r)!
● COMBINATION EXAMPLE
C (4, 3) = n!
r!(n-r)!
P (4, 3) = 4!
3!(4-3)!
P (4, 3) = 4!
3! 1!
P (4, 3) = 4.3!
3! 1
Simplify
P (4, 3) = 4
1
Simplify
P (4, 3) = 4
➔ Evaluate C(4, 3)

18. ● COMBINATION EXAMPLE
➔ In how many ways are there to chose 4 piece
of cards from 8 card deck?
COMBINATION Formula
C(n,r) = n!
r!(n-r)!
C (8, 4) = n!
r!(n-r)!
P (8, 4) = 8!
4!(8-4)!
P (8, 4) = 8!
4! 4!
P (8, 4) = 8.7.6.5.4!
4! 4.3.2.1
Simplify
P (8, 4) = 8.7.6.5.4!
4! 4.3.2.1
Simplify
P (8, 4) = 70
1
2
P (8, 4) = 70 ways

19. Thank you!!!