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Questions and Answers
What is the fundamental concept behind the Sum Rule Principle?
If two events A and B are disjoint, what is the value of n(A ∩ B)?
Suppose an event can occur in 5 ways, and independently of this, a second event can occur in 3 ways. How many ways can the events occur together?
What is the condition for the Sum Rule Principle to be applied?
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If there are 5 faculty members and 3 students, how many ways can a representative be chosen for a university committee if either a faculty member or a student can be chosen?
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What is the formula for the number of ways two events A and B can occur together, if they are independent?
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If two events A and B are not disjoint, what is the value of n(A ∪ B)?
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What is the main difference between the Sum Rule Principle and the Product Rule Principle?
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If there are 3 ways to choose a menu for a dinner party, and independently of this, there are 4 ways to choose a dessert, how many ways can the dinner party be planned?
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What is the purpose of combinatorial analysis?
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Study Notes
Product Rule Principle
- If two events occur independently, then the total number of ways they can occur is the product of the number of ways each event can occur.
- Formula: 𝑛( 𝐴× 𝐵 ) = 𝑛( 𝐴 ) × 𝑛( 𝐵 )
- Example: 3 flights from California to France and 2 flights from France to Sri Lanka, total number of flight plans is 3 × 2 = 6.
Sum Rule Principle
- If two events cannot occur simultaneously, then the total number of ways they can occur is the sum of the number of ways each event can occur.
- Formula: 𝑛( 𝐴+ 𝐵 ) = 𝑛( 𝐴 ) + 𝑛( 𝐵 )
- Example: 27 faculty members and 83 CS majors, total number of choices for a representative is 27 + 83 = 110.
Permutations
- An arrangement of items in a particular order, where order matters.
- Formula: 𝑛! / (𝑛−𝑟)!
- Example: 10 people in a competition, top three can be ordered in 10 × 9 × 8 = 720 ways.
- Permutations can be calculated using the fundamental counting principle or factorial notation.
Permutation Examples
- 3 people can be arranged in 3 × 2 × 1 = 6 ways.
- 10 people can be arranged in 10! / (10-3)! = 720 ways.
Permutation Practice
- A lock can be opened in 30 × 29 × 28 = 24360 ways.
- A club of 24 members can elect officers in 24 × 23 × 22 × 21 = 255,024 ways.
Combinatorial Analysis
- Includes the study of permutations and combinations.
- Concerned with determining the number of logical possibilities of some event without identifying every case.
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Description
Learn about the product rule principle in probability theory, including examples and formulas to calculate the number of ways events can occur independently.
