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Chapter 1: Problem 3

A tennis camp has 39 players. There are 25 left-handed players and 22 players who have a two-handed back stroke. How many left-handed players have a two- handed back stroke if every player is represented in these two counts?

Short Answer

Expert verified
8 left-handed players have a two-handed back stroke.

Step by step solution

01

Understand the Problem Context

We are given two groups within a tennis camp: left-handed players and players with a two-handed back stroke. We need to find out how many players are in both groups, i.e., left-handed players with a two-handed back stroke.
02

Apply the Principle of Inclusion-Exclusion

The Principle of Inclusion-Exclusion helps us calculate the overlap between two groups. The formula is: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] where \( n(A \cup B) \) is the total number of players, \( n(A) \) is the number of left-handed players, \( n(B) \) is the number of players with a two-handed back stroke, and \( n(A \cap B) \) is the overlap we need to find.
03

Plug in the Numbers

Using the formula from Step 2, substitute the known values: \[ 39 = 25 + 22 - n(A \cap B) \]. This represents the total number of players accounting for the overlap.
04

Solve for the Overlap

Rearrange the equation from Step 3 to solve for the overlap: \[ n(A \cap B) = 25 + 22 - 39 \]. Calculate: \[ n(A \cap B) = 47 - 39 = 8 \]. So, 8 left-handed players have a two-handed back stroke.

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Key Concepts

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

Set Theory
Set theory is a branch of mathematical logic that deals with collections of objects, known as sets. Sets are fundamental in mathematics for organizing and analyzing groups of related items. In the context of our problem, the sets we are dealing with are:
  • The set of left-handed players.
  • The set of players with a two-handed back stroke.
Each player in the tennis camp is a member of at least one of these sets, and some players might belong to both sets. When considering these sets, we often use notation such as:
  • \( n(A) \) for the left-handed players' set.
  • \( n(B) \) for the set of players with a two-handed back stroke.
  • \( n(A \cap B) \) to denote players who belong to both sets.
Set theory provides the concepts and vocabulary to describe relationships between these groups of players and allows us to analyze their intersections and overlaps effectively.
Counting Problems
Counting problems often involve determining the number of ways events can occur, using logic and reasoning to organize the counting process. For the exercise at hand, we are tasked with determining how many players belong to both sets: left-handed players and players with a two-handed back stroke. In solving counting problems, it is crucial to understand how to account for overlaps and intersections within sets. This is where the Principle of Inclusion-Exclusion (PIE) becomes essential. The basic inclusion-exclusion principle suggests that we first add up the total elements in each set and then subtract those that are counted twice, as they belong to the intersection of both sets. By structuring the problem using PIE, we can efficiently handle the double-counting in overlapping categories and find the desired count of players in both the left-handed and two-handed back stroke groups.
Discrete Mathematics
Discrete mathematics studies structures that are fundamentally discrete rather than continuous. It plays a key role in computer science, cryptography, and network theory, among others. In our context, we are looking at a discrete group of tennis players, and we want to apply mathematical principles to count without omissions or repetitions.The Principle of Inclusion-Exclusion is part of discrete mathematics. It allows us to solve complex counting problems by providing a formula to find overlaps in distinct groups:\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]This principle helps organize the information and compute the overlap in an efficient manner, making it a powerful tool in discrete mathematical problem-solving. By understanding and applying these principles, we can solve counting problems efficiently and accurately in various discrete settings, just like in our tennis camp problem.

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Most popular questions from this chapter

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