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Questions and Answers
If you have 4 different shirts and 3 different pants, how many unique outfits can you create following the fundamental counting principle?
In a restaurant, there are 4 appetizers and 5 main dishes. How many different ways can a customer choose one appetizer and one main dish following the fundamental counting principle?
If you have 2 types of fruit and 3 types of snacks, how many different snack-fruit pairs can you form according to the fundamental counting principle?
A store offers 3 different sizes of shirts and 4 different colors. How many unique combinations of shirt size and color can a customer choose?
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If a car is available in 5 colors and with 4 optional features, how many different configurations can be chosen for the car?
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In a game, there are 4 player positions and 6 available numbers to choose. How many different combinations can be formed by selecting one player position and one number?
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Study Notes
Fundamental Counting Principle
The fundamental counting principle is one of the most basic principles used in combinatorial counting. It's often referred to by its acronym, FCP. This principle applies when counting distinct objects where there may be overlapping categories. For instance, if you have three desks and each desk has two drawers, how many total drawers do you have? In this case, you would multiply the number of desks by the number of drawers each desk has, giving you (3 \times 2 = 6) total drawers.
This concept can also be applied to problems involving sets with elements that must be chosen from multiple sources; it's a powerful tool for determining the number of ways combinations can occur. Let's say you have ten numbers to choose from: (1, 2, 3,\ldots, 8, 9, 10.) If you were to choose three of them without replacement, meaning once a number is picked, it cannot be chosen again, you could count the possibilities using multiplication:
[3! = 3 \times 2 \times 1 = 6]
So, there are six different combinations possible when choosing three numbers out of ten without repetition.
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Description
Learn about the fundamental counting principle (FCP) and how it can be used to determine the total number of outcomes in scenarios involving distinct objects and combinations. Understand the concept through examples like counting the total number of drawers in desks or calculating different combinations possible when choosing elements from a set without replacement.
