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Chapter 16: Problem 32

Bob has 20 different dress shirts in his wardrobe. (a) In how many ways can Bob select seven shirts to pack for a business trip? (b) In how many ways can Bob select 5 of the 7 dress shirts he packed for the business trip \(-\) one for the Monday meeting, one for the Tuesday dinner, one for the Wednesday party, one for the Thursday conference, and one for the Friday date?

Short Answer

Expert verified
(a) The number of ways Bob can select 7 shirts from 20 is 77520. (b) The number of ways Bob can arrange 5 shirts from the 7 selected shirts is 2520.

Step by step solution

01

Calculate Number of Ways to Select Shirts

The number of ways Bob can select 7 shirts from 20 shirts is calculated using Combination formula: \( C(n, r) = n! / [(n-r)! r!] \) where n = total number of shirts (20) and r = number of shirts to select (7). So, we will substitute these values to get the number of combinations.
02

Calculate Number of Ways to Arrange Shirts

The number of ways Bob can arrange the 5 shirts from the 7 shirts selected is calculated using Permutation formula: \( P(n, r) = n! / (n-r)! \) where n = total number of shirts selected (7) and r = number of shirts to wear (5). So, we will substitute these values to get the number of permutations.
03

Evaluate Results

We should now evaluate the results obtained in Step 1 and Step 2 using the factorial formulas. The factorial of a number n is the product of all positive integers less than or equal to n. It's denoted as n!. After calculating the factorials, we will substitute them into the Combination and Permutation formulas.

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Key Concepts

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

Factorial
The concept of a factorial is fundamental in combinatorics. A factorial, denoted as \( n! \), is the product of all positive integers up to a number \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This concept is crucial when we delve into permutations and combinations, as it helps to calculate the total number of possible arrangements or selections.
Understanding that each factorial builds on the previous number aids in grasping more complex mathematical concepts.
Here are a few key points about factorials:
  • Factorials grow extremely fast; for instance, \( 7! = 5040 \), while \( 8! = 40320 \).
  • \( 0! \) is defined to be \( 1 \).
  • Factorials are only defined for non-negative integers.
Knowing these properties can help simplify calculations and prevent errors when we solve problems related to combinations and permutations.
Permutations
Permutations refer to the different ways of arranging a set of items where the order matters. If Bob has 5 shirts and wants to assign each one to a specific day, the arrangement on each day is a permutation problem.
The permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \) calculates the number of ways to rearrange \( r \) items from a set of \( n \) items. When calculating permutations:
  • Think about the number of possible orders.
  • Order is crucial, meaning \( ABC \) is different from \( BAC \).
  • For Bob, with 7 shirts choosing 5 for specific days, substitute \( n = 7 \) and \( r = 5 \) into the formula to find the permutations.
Using this formula, you account for all possible orderings and ensure accurate calculation of how items can be uniquely distributed.
Combinations
In contrast to permutations, combinations consider the selection of items where order does not matter. When Bob picks 7 shirts out of 20 for a trip, it doesn't matter which shirt he picks first, only which combination is used.
Use the combination formula \( C(n, r) = \frac{n!}{(n-r)! r!} \) to determine the number of possible combinations. Here's how combinations work:
  • Order of selection doesn't change the value.
  • Useful in choosing sets where arrangement afterwards isn't a factor.
  • For picking 7 shirts from 20, substitute \( n = 20 \) and \( r = 7 \) into the formula to find the combinations.
This culminates in calculating how many distinct ways a set group of items can be selected without regard for sequence.
Problem Solving in Mathematics
Problem-solving in mathematics can sometimes feel challenging. However, by breaking down a problem into manageable parts, like using factorials, permutations, and combinations, solutions become more accessible.
Here are some tips when approaching mathematical problems:
  • Understand the problem: Identify what is being asked.
  • Decide which formula to use: Is it a permutation or combination problem?
  • Solve using step-by-step calculations: Follow through with each part of the problem.
By practicing these steps, you become more confident in tackling mathematical problems. Realizing that solutions often require thinking through different methods helps build problem-solving skills and resilience.

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Most popular questions from this chapter

A box contains twenty \(\$ 1\) bills, ten \(\$ 5\) bills, five \(\$ 10\) bills, four \(\$ 20\) bills, and one \(\$ 100\) bill. You blindly reach into the box and draw a bill at random. What is the expected value of your draw?

An urn contains seven red balls and three blue balls. (a) If three balls are selected all at once, what is the probability that two are blue and one is red? (b) If three balls are selected by pulling out a ball, noting its color, and putting it back in the urn before the next selection, what is the probability that only the first and third balls drawn are blue? (c) If three balls are selected one at a time without putting them back in the urn, what is the probability that only the first and third balls drawn are blue?

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear. A die is rolled four times in a row. The observation is the number that comes up on each roll. Describe the sample space.

Two teams (call them \(X\) and \(Y\) ) play against each other in the World Series. (The World Series is a best-of-seven series: The first team to win four games wins the series, and games cannot end in a tic.) We can describe an outcome for the World Series by writing a string of letters that indicate (in order) the winner of each game. For example, the string \(X Y X X Y X\) represents the outcome: \(X\) wins game \(1, Y\) wins game \(2, X\) wins game \(3,\) and so on. (a) Describe the event \(" X\) wins in five games" (b) Describe the event "the series is over in game \(5 . "\) (c) Describe the event "the series is over in game 6 " (d) Find the size of the sample space.

The service history of the Prego SUV is as follows: \(50 \%\) will need no repairs during the first year, \(35 \%\) will have repair costs of \(\$ 500\) during the first year, \(12 \%\) will have repair costs of \(\$ 1500\) during the first year, and the remaining SUVs (the real lemons) will have repair costs of \(\$ 4000\) during their first year. Determine the price that the insurance company should charge for a one-year extended warranty on a Prego SUV if it wants to make an average profit of \(\$ 50\) per policy.
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