/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Three different awards are to be... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 15

Three different awards are to be given to a class of 15 students. Each student can receive at most one award. Count the number of ways these awards can be given out.

Short Answer

Expert verified
There are 2730 ways to distribute the awards.

Step by step solution

01

Identify the Problem Type

We are dealing with a counting problem where we need to determine how many ways we can distribute three distinct awards to a group of students where each student gets at most one award. This is a permutation problem because the awards are distinct and order matters.
02

Calculate the First Choice

For the first award, we can choose any one of the 15 students in the class to receive the award. This gives us 15 possible choices.
03

Calculate the Second Choice

After the first student is given an award, 14 students remain without an award. Therefore, for the second award, we have 14 choices.
04

Calculate the Third Choice

Once the first two awards have been given, 13 students remain without an award. Hence, we have 13 choices for the third award.
05

Compute the Total Number of Permutations

To find the total number of ways to distribute the awards, multiply the number of choices for each award: \[15 \times 14 \times 13 = 2730\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Counting Problems
Counting problems are a fundamental aspect of mathematics, specifically when dealing with questions that require finding the number of ways something can be arranged or organized. In the context of awards distribution, counting problems help identify how different arrangements can occur given specific conditions.
  • It's essential to determine whether order matters; if it does, we're dealing with permutations.
  • Counting helps address problems involving limited quantities or conditions, such as each student receiving at most one award.
  • Breaking the problem into smaller steps helps simplify complex counting tasks, as seen with awards.
Understanding counting problems can often involve recognizing patterns or utilizing mathematical tools, such as factorials or combinations, to calculate total possibilities.
Distinct Awards
When dealing with awards that are distinct, each award is unique and different from the others. This means that the order in which awards are given out matters significantly.
  • For instance, giving Awards A, B, and C to students Jane, John, and Joe would be a different arrangement than giving A to John, B to Joe, and C to Jane.
  • This distinct characteristic leads us to use permutation methods, as opposed to combinations, where order isn't important.
  • Identifying awards as distinct helps guide the approach for calculating the number of possible distributions.
By understanding that the awards are distinct, we apply suitable mathematical techniques to solve the problem effectively, ensuring we capture all possible outcomes.
Combinatorics
Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination within a set of elements. It is a powerful tool used in calculating possible arrangements where restrictions are involved.
  • In our problem, combinatorics helps assess how to distribute three distinct awards among 15 students while respecting the restriction that one award goes to one student only.
  • The concept of a permutation arises due to the distinct nature of these awards and the importance of order.
  • Utilizing combinatorics provides a systematic approach to what might seem like a daunting problem by breaking it into steps (using permutations) for clarity.
Combinatorics not only simplifies such counting problems but also enhances understanding of mathematical logic, paving the way for reasoning through similar challenges.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the Poisson approximation. Fragile \(X\) Syndrome About 1 in 1000 boys is affected by fragile \(X\) syndrome, a genetic disorder that causes learning difficulties. Find the probability that, in a group of 500 boys, nobody is affected by this disorder by (a) computing the exact probability and (b) using a Poisson approximation.

Prove Markov's inequality when \(X\) is a nonnegative discrete random variable with \(E(X)<\infty\)

The following data represent blood cholesterol levels, in \(\mathrm{mg} / \mathrm{dL}\), of patients in a clinical trial: $$ 174,138,212,203,194,245,146,149,164,209,158 $$ Find the median, the sample mean, and the sample variance.

Suppose \(X\) is Poisson distributed with parameter \(\lambda=2\). Find the probability that \(X\) is at least \(2 .\)

Suppose \(X\) is Poisson distributed with parameter \(\lambda=0.6\). Find the probability that \(X\) is less than \(3 .\)
See all solutions

Recommended explanations on Biology Textbooks

Biology Experiments

Read Explanation

Control of Gene Expression

Read Explanation

Ecosystems

Read Explanation

Cells

Read Explanation

Heredity

Read Explanation

Cell Communication

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.