Fundamental Counting Principle

Pauline Scripture:We present ourselves in the

name of Christ as if God Himself makes an appeal to you through us. Let God reconcile you, this we ask to you in the name of Christ… We appear to be afflicted, but we remain happy, we seem to be poor but we enrich many, apparently we have nothing, but we possess everything. 2 corinthians

5:20, 6:10

Suppose you decide to go to a gathering with your friends at a friend’s house. You find in

your closet 2 pairs of pants and

3 blouses. Assuming that each pair of pants can be matched

with any of the 3 blouses, how many ways can you dress for the gathering?

PANTS

BLOUSES

There are 6 ways to dress up.

Now, suppose you want to carry a purse

and you have 4 purses to choose

from.

The Multiplication Principle of Counting

If there are n1 ways to do the first task, n2 ways to do the second task, n3 ways to do the third task, and so on, then the total number of ways to perform the procedure is

n1 · n2 · n3 …

In how many ways can 3 coins fall?

Example 1

Answer 1

2 · 2 · 2 = 8

Example 2

A coin and a die are tossed. Then a card is picked from a standard deck. How many results are possible?

Answer 2

2 · 6 · 52 = 624

A high school class consist of 8

male and 6 female students. Find the number of ways that the class can elect

Example 3

a. two spokesperson (a male and a female)

b. a president and a vice president.

a. 8 · 6 = 48

Answer 3

b. 14 · 13 = 182

Example 4

In how many ways can 3 boys

and 2 girls sit in a row

a. if they can sit anywhere

b. if the boys and girls are to sit together

Answer 4

a. 5 · 4 · 3 · 2 · 1 = 120

b. 2(3 · 2 · 1 · 2 · 1) = 24

These can also be expressed asa. 5! = 120

b. 2(3! · 2!) = 24

Example 5A security code consists of two letters and three digits. How many distinct security codes are possiblea. if repetition is not allowed b. if the first character on the code is a vowel and repetition is not allowed?

a. 26 · 25 · 10 · 9 · 8

= 468 000b. 5 · 25 · 10 · 9 · 8

= 90 000

Answer 5

Work with a Partner

How many numbers of at

least 3-different digits can be formed from the

integers 1,2,3,4,5,6?

Answer:

a. form 3-different digit numbers

= 6 · 5 · 4 = 120b. form the 4-different digit numbers

= 6 · 5 · 4 · 3 = 360c. form 5-different digit numbers

= 6 · 5 · 4 · 3 · 2 = 720d. form 6-different digit numbers

= 6 · 5 · 4 · 3 · 2 · 1 = 720

= 120 + 360 + 720 +

720 = 1920

The Addition Principle of Counting

Suppose task 1 can be done in n1 ways, task 2 in n2 task 3 in n3, and so on (for a finite number of tasks). The total number of ways of doing task 1 or task 2 or task 3, and so on, is

n1 + n2 + n3 …

From a standard deck of 52 cards,

how many ways can we choose?a. a king or a queen?b. a heart, a diamond or a club?c. an even number or a spade?

Work with a Partner

Answer:

a. 4 + 4 = 8b. 13+13+13 =

39c. (4 · 5) + (13 – 5)

= 28

Factorialn! read as “n-factorial” is defined as for a positive integer n as:

n! = n (n-1) (n-2) (n-3) …

(2) (1)

Oral Drill:1.7!2.3! + 5!3.3!5!4.5!/4!5.(3!2!)/5!

Seatwork: Notebook1. How many different ways are there to

arrange the letters in the word SWITZERLAND?

2. How many 4 digit numbers can be formed from the digits 0 – 9:

a. if repetition is not allowed?b. the last digit must be zero and

repetitions are not allowed?c. if the numbers must be odd?d. if the numbers must be greater

than 6000 and repetitions are allowed?

3. How many 7 digits telephone numbers can be put on the 433 area code ifa. there are no repetitions?b. no telephone number can end

in 5?4. In how many ways can a 5

question multiple choice test be answered if there are four possible choices for each question?

5. How many odd integers between 2000 and 7000 have no repeated digits?

6. If repetitions are allowed, how many 3 digit numbers can be formed from the digits 0 – 5? How many of the numbers are odd? How many are even? How many are greater than 400?