Question

Speech Contest The number of ways that the first-, second-, and third-place winners in a speech contest can be selected from $n$ participants is given by$P(n)=n^{3}-3 n^{2}+2 n$. If there are seven participants, how many ways can the winner and first and second runners-up be selected?

Name: Problem 82, Chapter 5: Intermediate Algebra for College Students
Uploaded: 2020-03-14T01:07:17-08:00
Duration: 1 min 24 s
Channel: Heather Zimmers
Description: Speech Contest The number of ways that the first-, second-, and third-place winners in a speech contest can be selected from n participants is given byP(n)=n^3-3 n^2+2 n. If there are seven participants, how many ways can the winner and first and second runners-up be selected?

   Speech Contest The number of ways that the first-, second-, and third-place winners in a speech contest can be selected from $n$ participants is given by$P(n)=n^{3}-3 n^{2}+2 n$. If there are seven participants, how many ways can the winner and first and second runners-up be selected?

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 5, Problem 82

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Solved by Expert Heather Zimmers

03/14/2020

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There are 7 participants, so there are 7 ways to choose the winner. Show more…

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Speech Contest The number of ways that the first-, second-, and third-place winners in a speech contest can be selected from $n$ participants is given by$P(n)=n^{3}-3 n^{2}+2 n$. If there are seven participants, how many ways can the winner and first and second runners-up be selected?

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Key Concepts

Permutations

Permutations refer to the number of ways to arrange a set of items where the order matters. In problems like selecting ranked winners, each different ordering yields a distinct outcome, and the total count is often computed using permutation formulas.

Fundamental Counting Principle

The Fundamental Counting Principle states that if there are multiple tasks to be performed in sequence, the total number of outcomes is the product of the number of ways each individual task can be performed. This principle is used in problems where multiple positions are filled one after the other, each with a decrementally smaller set of candidates.

Ordered Selection

Ordered Selection is the process of choosing items from a set when the order in which they are selected is important. In contests and competitions, the positions (like winner, runner-up, etc.) are distinct, making the arrangement crucial to the final count of possible outcomes.

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