Permutations and Combinations

Let's review what we already know about the counting principle, permutations,

and combination.

Fundamental Principle of Counting: (also known as the multiplication rule

for counting) If a task can be performed in n1 ways, and for each of these a second task can

be performed in n2 ways, and for each of the latter a third task can be performed in n3 ways,

..., and for each of the latter a kth task can be performed in nk ways, then the entire sequence

of k tasks can be performed in n1 • n2 • n3 • ... • nk ways.

Permutation: A set of objects in which

position (or order) is important. To a permutation, the trio of Brittany, Alan and Greg is DIFFERENT

from Greg, Brittany and Alan. Permutations are persnickety (picky).

Combination: A set of objects in which

position (or order) is NOT important. To a combination, the trio of Brittany, Alan and Greg is THE SAME

AS Greg, Brittany and Alan.

Let's look at which is which:

Permutation versus Combination

1.Piking a team captain, pitcher, and shortstop

from a group.

1. Picking three team members from a group.

2. Picking your favorite two colors, in order,

from a color brochure.

2. Picking two colors from a color brochure.

3. Picking first, second and third place winners. 3. Picking three winners.

Formulas:

A permutation is the choice of r things from a set of n

things without replacement and where the order matters.

Special Cases:

A combination is the choice of r things from a set of n

things without replacement and where order does not

matter. (Notice the two forms of notation.)

Special Cases:

The term "combination" lock is mathematically confusing. To open such a lock, the "order" of the digits entered IS very important, unlike a

mathematical combination.

New Name: Permutation Lock

Example 1:

Evaluate :

Example 2: Joleen is on a shopping spree. She buys six tops, three shorts and 4 pairs

of sandals. How many different outfits consisting of a top, shorts and sandals can she create

from her new purchases?

Example 3: What is the total number of possible 4-letter arrangements of the letters

m, a, t, h, if each letter is used only once in each arrangement?

Example 4: There are 12 boys and 14 girls in Mrs. Schultzkie's math class. Find the number of ways

Mrs. Schultzkie can select a team of 3 students from the class to work on a group project.

The team is to consist of 1 girl and 2 boys.

Order, or position, is not important. Use the fundamental counting principle,

Practice (Guided 0dd # and Independent even #)

1. Determine whether the following situations would require calculating a permutation

or a combination:

a.) Selecting three students to attend a conference in Washington, D.C.

permutation combination

b.) Selecting a lead and an understudy for a school play.

permutation combination

c.) Assigning students to their seats on the first day of school.

permutation combination

2. Evaluate:

Choose:

12

60

480

720

3. A teacher is making a multiple choice quiz. She

wants to give each student the same questions,

but have each student's questions appear in a

different order. If there are twenty-seven

students in the class, what is the least number of

questions the quiz must contain?

4. Which of the following is NOT equivalent to ?

Choose:

5. A coach must choose five starters from a team of 12

players. How many different ways can the coach choose

the starters?

Answer

6. The local Family Restaurant has a daily breakfast special in which the customer

may choose one item from each of the following groups:

Breakfast Sandwich Accompaniments Juice

egg and ham breakfast potatoes orange

egg and bacon egg and cheese

apple slices fresh fruit cup

pastry

cranberry tomato apple grape

a.) How many different breakfast specials are possible?

b.) How many different breakfast specials without meat are possible?

Answer

7. In how many ways can 3 different vases be arranged on

a tray?

Answer

8. There are fourteen juniors and twenty-three seniors in

the Service Club. The club is to send four

representatives to the State Conference.

a.) How many different ways are there to select a group

of four students to attend the conference?

b.) If the members of the club decide to send two

juniors and two seniors, how many different groupings

are possible?

Answer

9. If represents the number of combinations of n items taken r at a time,

what is the value of when n=4?

Choose:

24 14 6 4

Homework

1. A locker combination system uses three digits from 0 to 9.

How many different three-digit combinations with no digit

repeated are possible?

Choose:

30

504

720

1,000

2. A fair coin is tossed three times. What is the probability

that the coin will land tails up on the second toss?

Choose:

1/3

1/2

2/3

3/4

3. A square dartboard is represented on the

accompanying diagram. The entire

dartboard is the first quadrant from x = 0

to 6 and from y = 0 to 6. A triangular

region on the dartboard is enclosed by

the graphs of the equations y = 2, x = 6,

and y = x. Find the probability that a

dart that randomly hits the dartboard will

land in the triangular region formed by

the three lines.

Answer

4. A bag contains 12 red M&Ms, 12 blue M&Ms, and 12

green M&Ms. What is the probability of drawing two

M&Ms of the same color in a row? When the first M&M

is drawn, it is looked at and eaten.

(HINT: This is a conditional probability problem.)

Choose:

11/35 12/36 11/105 2/36

5. A 10 x 20 foot mural, depicted below, shows a triangularly shaped region at the

bottom of the mural. Find the probability that a point selected at random will lie in

this triangular region of the mural.

Answer

6. The telephone company has run out of seven-digit

telephone numbers for an area code. To fix this

problem, the telephone company will introduce a new

area code. Find the number of new seven-digit

telephone numbers that will be generated for the new

area code if both of the following conditions must be

met:

• the first digit cannot be a zero or a one

• the first three digits cannot be the emergency

number (911) or the number used for

information (411).

7. Alex's wallet contains four $1 bills, three $5 bills, and

one $10 bill. If Alex randomly removes two bills

without replacement, determine whether the probability

that the bills will total $15 is greater than the probability

that the bills will total $2.

Answer

8. The party registration of the voters in Jonesville is

shown in the table. If one of the registered Jonesville

voters is selected at random, what is the probability

that the person selected in not a Democrat?

Answer

Registered Voters in Jonesville

Party

Number of voters

registered

Democrat 6,000

Republican 5,300

Independent 3,700

9. A bag of cookies contains 6 chocolate chip cookies, 5 peanut

butter cookies, and 1 oatmeal cookie. Brandon selects 2

cookies at random. Find the probability that Brandon selected :

a) 2 chocolate chip cookies

Answer

b) 1 chocolate chip cookie and 1 peanut butter cookie

Answer

10. A bag contains three chocolate, four sugar, and five

lemon cookies. Greg takes two cookies from the bag,

at random, for a snack. Find the probability that Greg

did not take two chocolate cookies from the bag.

Explain why using the complement of the event of

not choosing two chocolate cookies might be an

easier approach to solving this problem.

Answer

3. If there were two questions on the quiz, we could prepare two quizzes with the questions in different

order -- 2•1 = 2.

If there were three questions, we could get 3•2•1 = 6 different orders.

If there were four questions, we could get 4•3•2•1 = 24 different orders -- not quite enough for the class of 27

students.

If there were five questions, we could get 5•4•3•2•1 = 120 different orders. The teacher will need at least 5

questions on the quiz.

5. Choose 5 starters from a team of 12 players. Order is not important.

6. a.) Basic counting principle:

Sandwiches x Accompaniments x Juice

3 • 4 • 5 = 60 breakfast choices

b.) Meatless means that under Sandwiches there will be only one choice.

Sandwiches x Accompaniments x Juice

1 • 4 • 5 = 20 meatless breakfast choices

8. 14 juniors, 23 seniors

37 students total

a.) Choose 4 students from the total number of students. Order is not important.

b.) Choose 2 juniors and 2 seniors.

HW (key)

3. The intersecting lines form a right triangle. The area of

the triangle is (0.5)(4)(4) = 8 square units. The area of

the square dartboard is (6)(6) = 36. The probability that

the dart will hit the triangular region is 8/36 or 2/9 or 2:9.

5. The area of the entire mural is (10)(20) = 200 square

feet.

The area of the triangle is (0.5)(20)(8) = 80 square feet.

The probability of choosing a point in the triangular

region is 80/200 or 4/10 or 2/5 or 2:5.

6. If picking a seven digit number from the 10 digits of 0 -

9, with repetition of digits, we would get

.

But we must not allow 0 or 1 for the first digit, so now

we have .

But we must also not allow 911 (911-XXXX) or 411

(411-XXXX), so we subtract out .

ANSWER:

7. Four $1 bills, three $5 bills, and one $10 bill = a total of

8 bills.

If the 2 bills total $15, the bills are one $5 and one $10.

If the 2 bills total $2, the bills are two $1.

The probability of totaling $15: Be sure to consider

which bill is chosen first:

OR

The probability of totaling $2:

The probability of the bills totaling $15 is NOT greater

than the probability of the bills totaling $2.

8. Total voters = 15,000

Not Democrat total = 9,000

Probability of not being Democrat = 9/15 = 3/5 = 0.6

OR

Probability of Democrat = 6/15 = 2/5

Complement = 3/5 = 0.6

9.

a) Choose 2 chocolate chip How many cookies will be chocolate chip

when choosing 2 cookies from the total

number of cookies?

b) Choose 1 chocolate chip and 1 peanut

butter

10. Choosing to use the complement will limit the

number of possible combinations that will need to be

examined. If we know the probability of two

chocolate cookies, the complement will tell us the

probability of NOT chocolate. Otherwise, we will

need to consider ALL of the possible combinations

other than chocolate.

Probability of chocolate:

Probability of NOT chocolate: