The Fundamental Counting Principle - Room...
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Combinatorics Part One.notebook 1 December 01, 2015 Combinatorics • The Fundamental Counting Principle • Tree Diagrams • Factorial Notation Combinatorics is the mathematics of counting.
Transcript of The Fundamental Counting Principle - Room...
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Combinatorics
• The Fundamental Counting Principle• Tree Diagrams• Factorial Notation
Combinatorics is the mathematics of counting.
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The Fundamental Counting Principle(also known as the Multiplication Principle)
The Fundamental Counting Principle is a wayto figure out the total number of ways different events can occur.
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The Fundamental Counting Principal
If one task can be performed in a ways, a second task b ways, and a third task c ways, then all three tasks can be arranged in abc ways.
The counting principle can be extended to any number of tasks, providing the outcome of no one task influences the outcome of another.
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1. A restaurant offers 10 appetizers and 15 main courses. In how many ways can you order a twocourse meal?
Examples
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2. A pizza can be ordered with two choices of size (medium or large), three choices of crust (thin, thick or regular), and five choices of toppings (beef, sausage, pepperoni, bacon or mushroom). How many different onetopping pizzas can be ordered?
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3. You are taking a multiplechoice test that has six questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?
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4. How many telephone number are available with the 548 prefix?
Note: A prefix in a telephone number is followed by a 4 digit number. Each digit has a choice of 10 numbers, 0 to 9.
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5. How many ways can the letters in the word PENCIL be arranged?
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Tree Diagrams
A tree diagram is a very good way to visualize and count the total number of outcomes of an event.
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Examples
1. You are at a carnival. One of the carnival games asks you to pick a door and then pick a curtain behind the door. There are 3 doors and 4 curtains behind each door. How many choices are possible for the player?
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2. The 4 aces are removed from deck of cards. A coin is tossed and one of the aces is chosen. Draw a tree diagram to illustrate the number of possible outcomes.
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3. There are 3 trails leading to Camp A from your starting position. There are 3 trails from Camp A to Camp B. Draw a tree diagram to illustrate the number of routes from your starting position to Camp B.
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Factorial Notation
The product of the consecutive positive integers from 1 to n is given a special name, n factorial , which is written n!
Factorial notation is an abbreviation for products of
successive positive integers.
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Note
A factorial problem is a problem that involves counting the number of ways that a set of things can be arranged in different orders.
In general, to count the number of different ways that n things can be put in order, the answer is found by multiplying all of the integers from n down to 1.
n! can be used to represent the number of ways to arrange n distinct objects.
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Definition of n factorial (n!)
A factorial is defined, for nonnonegative integers, by:
Note: By definition, 0! = 1
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Examples
1. Simplify 6!
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3. Simplify
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4. Simplify
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5. Solve for n.
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6. Solve for n.
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7. How many different "words" can be made using all the letters in abcde?
There are five distinct letters to arrange.
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8. The manager of a softball team needs to put the 9 players on the team into a batting order. This means players have to be assigned places from 1st through 9th. How many different batting orders are possible?
There are 9 distinct players to arrange.
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There are 26 distinct letters to arrange.
9. How many ways can you arrange all the letters in the alphabet?
