This document discusses four counting techniques: systematic listing, tree diagrams, the fundamental counting principle (FCP), and permutations. It provides examples of using each technique to solve problems involving counting possible outcomes. FCP states that if there are m ways to do one thing and n ways to do another, there are m*n ways of doing both. Permutations use factorials to count arrangements of distinct objects where order matters. The document gives examples of using permutations to count arrangements of people in roles and letters in words.

1. four counting
techniques
prepared: ms. rigen v. maalam

2. review

3. systematic
listing
COUNTING TECHNIQUES
making a table
fundamental
counting principle
tree
diagram

4. Tree diagram
It uses line segments originating
from an event to an outcome
FCP
It states that "when there are m
ways to do one thing, and n
ways to do another, then there
are (m)(n) ways of doing both."

5. Sarah goes to her local pizza parlor
and orders a pizza. She can choose
either a large or a medium pizza, can
choose one of seven different
toppings, and can have three different
choices of crust. How many different
pizzas could Sarah order?
previous quiz

6. Sarah goes to her local pizza parlor
and orders a pizza. She can choose
either a large or a medium pizza, can
choose one of seven different
toppings, and can have three different
choices of crust. How many different
pizzas could Sarah order?
previous quiz

7. Pizza:
2 sizes, 7 toppings, 3 crusts
By FCP: (2)(7)(3) = 42
There 42 possible combinations
for the pizza
previous quiz

8. A person wants to buy one fountain
pen, one ball pen and one pencil from a
stationery shop. If there are 10 fountain
pen varieties, 12 ball pen varieties and
5 pencil varieties, in how many ways
can he select these articles?
your turn!

9. No. of ways in selecting fountain pen = 10
No. of ways in selecting ball pen = 12
No. of ways in selecting pencil = 5
Total number of selecting all these
= 10 x 12 x 5
= 600
sample PROBLEM

10. permutation of
distinct objects

11. It is a counting technique that determines the
number of possible arrangements in a collection of
items where the order of the selection matters.
permutation
The word 'distinct' means that the objects of a set
must be all different. There shall be no repetition of
occurrence.
distinct objects

12. Choosing a president, a vice president, a secretary and
a treasurer from the 12 members in a club
Arranging 3 different mathematics of books in a shelf
Getting the possible arrangements of letters of READ
permutation of distinct objects
EXAMPLES:
It refers to the different arrangements of distinct
objects in a line. It is the different arrangements when
no objects are identical or the same.

13. How many possible
arrangements could be made
when selecting a president, a
vice president, a secretary and a
treasurer from the 12 members
in a club
sample problem 1

14. There are 12 members.
No. of ways in selecting:
By FCP: (12)(11)(10)(9) = 11, 880
a. president: 12
b. vice-president: 11
c. secretary: 10
d. treasurer: 9
Thus, there are 11, 880 ways to arrange 12 members
of the club for the position P, VP, S andT.
SOLUTION

15. A librarian wanted to organize a
book shelf. In how many
arrangements can be made with
3 different mathematics books?
sample problem 2

16. There are 3 different mathematics books.
No. of ways in selecting:
By FCP: (3)(2)(1) = 6
a. 1st book: 3
b. 2nd book: 2
c. 3rd book: 1
Thus, there are 6 ways to arrange 3 different
mathematics books in a shelf.
SOLUTION

17. In how many ways can you
arrangement the letters of the
word READ?
sample problem 3

18. There are 4 different letters of the word READ
No. of ways in selecting:
By FCP: (4)(3)(2)(1) = 24
a. 1st letter: 4
b. 2nd letter: 3
c. 3rd letter: 2
d. 4th letter: 1
Thus, there are 24 ways to arrange the letters of the
word READ.
SOLUTION

19. RECALL
a. Sample Problem 2 solution: (3)(2)(1) = 6
b. Sample Problem 3 solution: (4)(3)(2)(1) = 24
It means to multiply all whole numbers from the
chosen number down to 1.
In symbols, " ! "
This operation is called Factorial.

20. Take note
3! = (3)(2)(1) = 6
4! = (4)(3)(2)(1) = 24
7! = (7)(6)(5)(4)(3)(2)(1) = 5040
Factorial (!) means to multiply all whole numbers from
the chosen number down to 1.
Examples:

21. calculator

22. Take note
Anagram is a word or phrase created by rearranging all
the letters of a certain word. The letters must be used
only once and the word that is formed has meaning.
Examples:
READ
DEAR
DARE

23. How many anagrams of the
word SHARE are there?
sample problem 4

24. answer
HARES
SHEAR
SHAER
HEARS
RHEAS -a large flightless bird resembling
a small ostrich
-to receive a sound
-to remove fleece or hair by clipping
or cutting
-a fast-running, long-eared mammal
that resembles a large rabbit
-a limited amount
There are 5 anagrams of the word SHARE.
Anagrams of SHARE are:

25. permutation of distinct objects
All objects are different.
No objects are identical
or the same
Can be solved using the
Fundamental Counting
Principle
Can be solved using
Factorial
May use the concept of
Anagrams

26. Answer the exercises on your worksheet.
You may use a calculator to solve for
factorial.
activity