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PRINCIPLES OF COUNTING
OBJECTIVES:
- Construct a Tree Diagram
- Apply the Multiplication Principle
- Use Factorial Notation
- Apply the Permutation Rule
- Apply the Combination Rule
OBJECTIVE 1: CONSTRUCT A TREE DIAGRAM
A tree diagram is helpful in visualizing all possible outcomes, choices, or events. Suppose an experiment consists of tossing a coin, followed by rolling a die. To list all the possible outcomes, we can construct a tree diagram.
All possible outcomes are H1, H2,... , and T6. There are 2 6 12 possible outcomes.
EXAMPLE 1 A restaurant has a lunch special menu. The lunch special will be served on one plate; which includes rice and two items. Item 1: Side: noodles, salad, or stir fry vegetable Item 2: Entrée: eggplant with tofu, broccoli chicken, pepper steak, or spicy beef
Construct a tree diagram and find the total number of possible plate combinations.
Solution:
H
1 H
2 H
3 H 4 H
5 H
6 H
T
1 T 2 T 3 T 4 T
5 T 6 T
H = Head T = Tail
All possible plate combinations are ne, nc, np, nb, se, sc, sp,.. ., vb. There are 3 4 12 plate combinations
for the lunch special.
OBJECTIVE 2: APPLY THE MULTIPLICATION PRINCIPLE:
EXAMPLE 1 You have to choose a password containing exactly 4 characters, i.e. 2 letters (case sensitive), followed by 2 numbers, how many different choices of passwords are possible?
Solution:
Possible choices Possible choices Possible choices Possible choices for the first letter for the first letter for the first number for the second number (a – z, A – Z) (a – z, A – Z) (0 – 9) (0 – 9)
52 52 10 10 = 270,
There are 270,400 choices of passwords.
noodles (n)
eggplant with tofu (e)
broccoli chicken (c)
pepper steak (p)
spicy beef (b)
salad (s)
eggplant with tofu (e)
broccoli chicken (c)
pepper steak (p)
spicy beef (b)
stir fry vegetable (v)
eggplant with tofu (e) broccoli chicken (c)
pepper steak (p)
spicy beef (b)
Multiplication Principle
If a choice consists of i steps, of which the first step can be made in n 1 ways, the second in n 2 ways,…, and the last step in ni ways, then the number of different ways possible choices can be made is
n 1 n 2 n 3 ni
OBJECTIVE 3: USE FACTORIAL NOTATION
EXAMPLE 1 Calculate 5!
Solution:
EXAMPLE 2
Calculate15!
Solution:
EXAMPLE 3 Find the number of ways we can arrange the letter A, B, C, D if no repetition is allowed.
Solution: We can use either the multiplication principle or factorial notation.
Using Multiplication Principle
Since no repetition is allowed, we only have 3 choices for the second letter. We use one choice for the first letter so only 3 choices are left for the second letter, 2 choices left for the third and 1 choice left for the last letter.
First Letter Second Letter Third Letter Fourth Letter (4 choices) (3 choices) (2 choices) (1 choice)
Using Factorial Notation
Factorial Notation yields the same result.
Factorial Notation
If n is a natural number, then n! (read as “ n factorial”), is given by
n n n n n
Therefore, there are 24 ways to arrange the letters A, B, C, and D if no repetition is allowed.
OBJECTIVE 4: APPLY THE PERMUTATION RULE
EXAMPLE 1 How many 3 digit numbers can be formed from numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 if no repetition is allowed?
Solution:
Understanding
Here we can use permutation with n = 9 and r = 3 because the order is important and there is no repetition. The number 215 is different from 125.
Using the Permutation Rule
9 3
P ^ ^
Interpreting the result
There are 504 different 3-digit numbers which can be formed from numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 if no repetition is allowed.
Note : We can also use the multiplication principle to answer this question.
EXAMPLE 2 A board of directors of an HOA (Home Owner’s Association) consists of 8 people. How many ways can the HOA members pick a president, a secretary, and a treasurer from the board of directors?
Solution:
Understanding We can use permutation with n = 8 and r = 3 because the order is important and there is no repetition.
Permutation Rule
The number of ways in which r objects can be selected in a specific order from n distinct objects
when order is important and no object is repeated is given by the permutation n Pr , where
n r ( )!
P n
n r
Note : n Pr is sometimes written as Pn r , and is read as “ n permute r ” or “ n select r ”.
Interpreting the Result There are 35 different color combinations containing three colors out of the seven primary colors of rainbow.
EXAMPLE 2 A committee consisting 12 members must form a subcommittee consisting of 4 members. How many different subcommittees are possible?
Understanding Here, we can use combination with n = 12 and r = 4. The order is not important because there is no ranking of subcommittee members. Also there is no repetition.
Using the Combination Rule
12 4
C ^ ^ ^
Interpreting the Result There are 495 possible subcommittees of size 4 chosen from the 12 members.
EXERCISES: Constructing a Tree Diagram
In exercises 1 - 2, construct a tree diagram to represent all possible choices (or outcomes) and determine the total number of available choices (or outcomes).
- A student goes to a coffee shop and has two choices for ordering coffee. Choice 1: Size: small, medium, or large Choice 2: Type of coffee drink: cappuccino, mocha, espresso, Americano, or Frappuccino
- A health survey consists of three questions. Question 1: Gender: male, female, or other Question 2: Age group: under 18, 18 – 30, 31 – 50, above 50 Question 3: Has the person smoked: yes or no
Applying the Multiplication Principle
In exercises 3 - 7, apply the multiplication principle to answer the questions.
- Tom’s Pizza offers three sizes of pizza: small, medium, and large. There are 2 types of crusts and 12 types of toppings. How many ways can a person order a pizza with one topping?
- How many different ways can we arrange the letters of the word PLANTS?
- A tourist in Spain wants to visit 8 cities in 3 weeks. How many different routes are possible?
- A luggage lock has 4 dials. Each dial has the digits 0 to 9. How many different combinations are there for the lock?
- License plates in California consist of one number followed by three letters and three numbers. How many different license plates can be made?
Using the Factorial Notation
In Exercises 8 - 13, calculate the given factorial.
Using the Permutation Rule
In Exercises 14 - 27, calculate the given permutation. Express large values using E-notation with the mantissa rounded to two decimals.
14. 7 P 2
15. 16 P 4
16. 24 P 4
17. 35 P 10
18. 6 P 6
19. 50 P 49
20. 17 P 15
21. 27 P 7
22. 8 P 7
- To play Powerball, a person needs to select five numbers from 1 to 69 for the white balls; then select one number from 1 to 26 for the red Powerball. How many different ways can the Powerball numbers be picked?
- The CEO of a company wants to visit the company’s branch offices in New Mexico, Texas, and Louisiana. The company has 10 offices in New Mexico, 12 in Texas, and 5 in Louisiana. If the CEO wants to pick 5 in both New Mexico and Texas, and 3 in Louisiana to visit, how many different ways can he pick the branch offices to visit?
- Arnold is going on a camping trip. He has 10 favorite shirts, but he plans to bring only 4 shirts. He has 5 different pairs of pants, but he wants to bring 2 pairs. How many different outfits can Arnold wear in his camping trip?
- A jar of Halloween treats contains of 24 pieces of Hershey’s chocolates and 12 pieces of Reese’s peanut butter cups. In how many ways can Anna pull out 3 pieces of chocolates and 2 pieces of peanut butter cups?
