Counting ProblemsCounting Problems

Multiplication Rule Combinations Permutations

Goals of Lesson

• Look at 3 special kinds of counting problems

• Connect the kind of counting problem to correct formula

• Use our calculators to solve counting problems

First kind of counting problem: Multiplication

Rule

• How many different arrangements can be made if–each item can be used more than once for each arrangement

Counting Area Codes• The phone company assigns three-

letter area codes to represent locations. For example, the area code for Northern California is 707. How many different 3 digit area codes are possible if you can use the same number more than once in an area code?

We will assume that any combination of three numbers is possible.

Multiplication Principle of Counting:

(# of choices for 1st position) (# of choices for 2nd position) (# of choices for 3rd position)……

for however many positions

In the area code problem there are 3 positions with 10 choices for each

• How many area codes are possible if we can use a number more than once in the same area code?

• Three positions: ____ ____ ____

• Available choices for each digit (10 total): • 0 1 2 3 4 5 6 7 8 9 • Number of choices for 1st position: 10• Number for 2nd position: 10• Number for 3rd position: 10

Total number of possible area codes

= (10)(10)(10)

= 1000

Second kind of counting problem: Permutation

• How many different arrangements can be made if– each item can only be used once for

each sequence – order matters (a b c not the same as c

b a)

Example of a Permutation Problem:If you can only use each letter once,

how many different arrangements can be made using the letters in the alphabet?

(There are 26 letters in the alphabet)

n Pr

P stands for Permutation

n is the number of objects we can pick from

r is the number of objects we use in each arrangement

How many ways can we arrange the letters

of the alphabet if we can only use each

letter once? (nPr)

P stands for Permutation

n = 26

r = 26

Calculator Emulator

26 P 26 = 4.03 x 1026

Example of another Permutation:

• Three members of a club with 20 students will be randomly selected to serve as president, vice president, and treasurer.

• The first person selected will be president.

• The second person selected will be vice president.

• The third person selected will be treasurer.

How many president-vice president-treasurer arrangements are possible?

20 P3 think of it as 20

people picked 3 at a time

How many president-vice president-treasurer combinations are possible?For these arrangements, we will only use 3 people at a time even though there are 20 people.

This means: If we have 20 people (number) and we randomly pick 3 at a time, there are 6840 president-vice president-treasurer combinations

Calculator Emulator

Betting on the Trifecta

• In how many ways can horses in a 10 horse race finish first, second, and third?

• Number of horses: 10• Randomly picked 3 at a time• Same horse can’t be 1st and 2nd at same

time so no repetition in same pick• Order matters: 1st place isn’t same as

2nd .

•10 P 3

720 ways that top 3 horses could finish 1st, 2nd, and 3rd

Third kind of counting problem: Combinations

• How many different arrangements can be made if– each item can be only be used once

for each arrangement– order does NOT matter

What if order doesn’t matter?

• In horse races, order matters– Hedgehunter, Royal Auclair, Simply

Gifted placed 1st, 2nd, and 3rd in the 158th Grand National

• The order they came in mattered.• In poker, order does NOT matter

– if you get 2 kings and 3 queens (full house), it doesn’t matter what order you hold them in your hand)

• In poker, the combination of cards matters, not the order.

Combinations

• Teresa, Bethany, Marissa, and Bridgette are going to play doubles tennis. They will randomly select teams of two players each. List all of the combinations possible.

Teresa-Bethany Teresa-Marissa

Teresa-Bridgette Bethany-Marissa

Bethany-Bridgette Marissa-Bridgette

Combinations

• A combination is different from a permutation.

• A combination is when order doesn’t matter

• In a combination, ABC is the same as BAC

Formula for Combinations

n C rn = number of choices

C stands for Combination

r = number in each arrangement

• Remember the 4 girls that formed teams of two for tennis?

• We wanted to know how many different teams of two could be formed, and order did not matter

n C r n = 4

r = 2 Calculator Emulator

4 C 2 = 6

• How many different teams of size 4 can be made if there are 20 people?

• 20 people n = 20 • randomly choose 4 at a time r =

4

20 C 4 = 4845

What was the first kind of counting problem we

looked at today?

1.Multiplication Rule Problems

Arranged a given number of objects

The same object could be used more than once in each arrangement

How many area codes can be formed with the numbers 0 through 9 if numbers can be repeated?

(10) (10) (10) = 1000

What was the second kind of counting

problem we looked at today?

2. Permutation Problems

Arrangement of objects

Could not use the same object more than once in an arrangement

Order mattered (ABC different from BCA)

In how many ways can horses in a 10 horse race finish 1st, 2nd, and 3rd?

n = 10 r = 3 10 P3 = 720

What was the third kind of counting

problem we looked at today?

3.Combination Problems

Counting how many ways to arrange objects

Could only use each object once for each arrangement

Order did not matter (ABC same as (BCA)How many different teams of two can be formed from 4 players?

n = 4 r = 2 4C2 = 6

If same object can be used more than once in the same arrangement, then

Multiplication Rule

If same object can NOT be used more than once in the same arrangement, and order matters, then

Permutation

If same object can NOT be used more than once in the same arrangement, and order DOESN’T matter, then

Combination