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Chapter 8: Problem 8

In a survey of 270 college students, it is found that 64 like Brussels sprouts, 94 like broccoli, 58 like cauliflower, 26 like both Brussels sprouts and broccoli, 28 like both Brussels sprouts and cauliflower, 22 like both broccoli and caulifower, and 14 like all three vegetables. How many of the 270 students do not like any of these vegetables?

Short Answer

Expert verified
116 students don't like any vegetables.

Step by step solution

01

- Understand the Problem

Identify how many students like each vegetable and how many like multiple vegetables. The problem involves 64 students liking Brussels sprouts (B), 94 students liking broccoli (R), 58 students liking cauliflower (C), 26 liking Brussels sprouts and broccoli (B 鈭� R), 28 liking Brussels sprouts and cauliflower (B 鈭� C), 22 liking broccoli and cauliflower (R 鈭� C), and 14 liking all three vegetables (B 鈭� R 鈭� C).
02

- Apply the Principle of Inclusion-Exclusion

Use the principle of inclusion-exclusion for three sets to find the number of students who like at least one of the vegetables. The inclusion-exclusion formula for three sets is: \[ |B \text{ or } R \text{ or } C| = |B| + |R| + |C| - |B 鈭� R| - |B 鈭� C| - |R 鈭� C| + |B 鈭� R 鈭� C| \]Substitute the given values into the formula: \[ |B \text{ or } R \text{ or } C| = 64 + 94 + 58 - 26 - 28 - 22 + 14 \]
03

- Perform the Calculations

Calculate the total number of students who like at least one of the vegetables: \[ |B \text{ or } R \text{ or } C| = 64 + 94 + 58 - 26 - 28 - 22 + 14 = 154 \]
04

- Determine Students Who Don't Like Any Vegetables

Subtract the number of students who like at least one vegetable from the total number of surveyed students (270): \[ 270 - 154 = 116 \]

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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 fundamental branch of mathematics used to group and analyze objects, called elements. In this problem, the 'objects' are students, and the 'sets' represent students who like different vegetables. Each set is defined by a specific characteristic (e.g., liking Brussels sprouts). For example:
  • Set B = students who like Brussels sprouts (64 students)
  • Set R = students who like broccoli (94 students)
  • Set C = students who like cauliflower (58 students)
Understanding these sets and their intersections (students who like multiple vegetables) helps us use mathematical principles to solve complex problems.
The inclusion-exclusion principle plays a key role in combining these sets accurately.
Survey Analysis
Survey analysis involves collecting data and using it to draw meaningful conclusions. In this exercise, we analyze survey data about students' vegetable preferences. It's important to correctly interpret the survey results to understand how many students like single, multiple, or none of the vegetables.
  • 26 students like both Brussels sprouts and broccoli (B 鈭� R)
  • 28 students like both Brussels sprouts and cauliflower (B 鈭� C)
  • 22 students like both broccoli and cauliflower (R 鈭� C)
  • 14 students like all three vegetables (B 鈭� R 鈭� C)
Analyzing this data helps in accurately applying mathematical formulas to find additional insights, such as the number of students who don't like any vegetables (116 in this case).
Combinatorics
Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of objects. In the context of this problem, we are counting how many students belong to combinations of different sets. The principle of inclusion-exclusion is a key combinatorial method here.
This principle helps to correctly count the number of students in the union of sets (students who like at least one vegetable) by adjusting for double and triple counting of students in intersections.
The formula for three sets is:
Substituting the given numbers:
|B 鈭� R 鈭� C| = 64 + 94 + 58 - 26 - 28 - 22 + 14 = 154
This adjustment ensures that all overlapped students are counted correctly.
Mathematical Problem-Solving
Mathematical problem-solving involves the application of various mathematical techniques to find solutions to given problems. In this case, we used the Principle of Inclusion-Exclusion to solve the problem step-by-step:
  • Identify the sets and their intersections.
  • Apply the appropriate formula to find the number of students who like at least one vegetable.
  • Perform the necessary calculations to avoid double or triple counting.
  • Subtract the result from the total number of surveyed students to find the number of students who do not like any of the vegetables (116 students).
Using a clear, structured approach makes solving even complex problems manageable and understandable.

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Most popular questions from this chapter

Exercises \(38-45\) involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and \(n\) disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg 1 to peg 4 so that no disk is ever on top of a smaller one. This algorithm, given the number of disks \(n\) as input, depends on a choice of an integer \(k\) with \(1 \leq k \leq n .\) When there is only one disk, move it from peg 1 to peg 4 and stop. For \(n>1\) , the algorithm proceeds recursively, using these three steps. Recursively move the stack of the \(n-k\) smallest disks from peg 1 to peg \(2,\) using all four pegs. Next move the stack of the \(k\) largest disks from peg 1 to peg \(4,\) using the three-peg algorithm from the Tower of Hanoi puzzle without using the peg holding the \(n-k\) smallest disks. Finally, recursively move the smallest \(n-k\) disks to peg \(4,\) using all four pegs. Frame and Stewart showed that to produce the fewest moves using their algorithm, \(k\) should be chosen to be the smallest integer such that \(n\) does not exceed \(t_{k}=k(k+1) / 2,\) the \(k\) th triangular number, that is, \(t_{k-1}

Use generating functions (and a computer algebra pack- age, if available) to find the number of ways to make change for \(\$ 1\) using pennies, nickels, dimes, and quarters with a) no more than 10 pennies. b) no more than 10 pennies and no more than 10 nickels. c) no more than 10 coins.

Find a closed form for the generating function for the se- quence \(\left\\{a_{n}\right\\},\) where a) \(a_{n}=5\) for all \(n=0,1,2, \ldots\) b) \(a_{n}=3^{n}\) for all \(n=0,1,2, \ldots\) c) \(a_{n}=2\) for \(n=3,4,5, \ldots\) and \(a_{0}=a_{1}=a_{2}=0\) d) \(a_{n}=2 n+3\) for all \(n=0,1,2, \ldots\)

Use the principle of inclusion-exclusion to derive a formula for \(\phi(n)\) when the prime factorization of \(n\) is $$n=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{m}^{a_{m}}$$

Suppose that the function \(f\) satisfies the recurrence relation \(f(n)=2 f(\sqrt{n})+\log n\) whenever \(n\) is a perfect square greater than 1 and \(f(2)=1\) . a) Find \(f(16)\) . b) Find a big- \(O\) estimate for \(f(n) .\) [Hint: Make the substitution \(m=\log n . ]\)
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