4. Probability
Fundamental Counting Principle
1:10 minutesProblem 4.4.1
Textbook Question
Notation What does the symbol ! represent? The five starting players of an NBA basketball team can stand in a line 5! different ways, so what is the actual number of ways that the five players can stand in a line?
Verified step by step guidance1
The symbol '!' represents a factorial in mathematics. The factorial of a number n, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1.
To calculate the number of ways the five starting players can stand in a line, we need to compute the factorial of 5, which is written as 5!.
Write out the factorial expression for 5!: 5! = 5 × 4 × 3 × 2 × 1.
Multiply the numbers step by step: Start with 5 × 4, then multiply the result by 3, then by 2, and finally by 1.
The result of this multiplication will give you the total number of ways the five players can stand in a line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factorial
The symbol '!' represents the factorial of a number, which is the product of all positive integers up to that number. For example, 5! (read as 'five factorial') equals 5 × 4 × 3 × 2 × 1, which calculates the total number of ways to arrange five distinct objects.
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Combinations
Permutations
Permutations refer to the different arrangements of a set of items where the order matters. In the context of the question, the arrangement of the five NBA players in a line is a permutation, as changing the order of players results in a different arrangement.
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Combinatorial Counting
Combinatorial counting is a branch of mathematics that deals with counting, arranging, and combining objects. It provides the foundational principles for calculating permutations and combinations, which are essential for solving problems involving arrangements, such as the number of ways players can line up.
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