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Chapter 11: Problem 29

Decide whether the situation described involves a permutation or a combination of objects. (a) a telephone number (b) a Social Security number (c) a hand of cards in poker (d) a committee of politicians (e) the "combination" on a padlock (f) an automobile license plate (g) a lottery choice of six numbers where order does not matter

Short Answer

Expert verified
a) permutation, b) permutation, c) combination, d) combination, e) permutation, f) permutation, g) combination

Step by step solution

01

- Understand Definitions

First, recall the definitions: Permutations are arrangements of objects where the order matters, while combinations involve selecting objects where the order does not matter.
02

- Analyze Each Situation

Evaluate each situation to determine if the order of objects is important or not.
03

Step 2a - Telephone Number

The order of digits in a telephone number is important. Therefore, it is a permutation.
04

Step 2b - Social Security Number

The order of digits in a Social Security number is also important. Hence, it is a permutation.
05

Step 2c - Hand of Cards in Poker

In poker, the order of the cards in a hand does not matter, only the combination of cards matters. Therefore, it is a combination.
06

Step 2d - Committee of Politicians

The order of selecting committee members does not matter. Hence, it is a combination.
07

Step 2e - 'Combination' on a Padlock

Despite the name, the order of numbers for a padlock combination matters strictly. Therefore, it is a permutation.
08

Step 2f - Automobile License Plate

The order of characters on an automobile license plate matters. Hence, it is a permutation.
09

Step 2g - Lottery Choice of Six Numbers

In this type of lottery, the order of the chosen numbers does not matter. Therefore, it is a combination.

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Key Concepts

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

Permutations
Permutations are all about order. When we think about arrangements where the sequence is crucial, we are talking about permutations. For example, a telephone number or a Social Security number are typical permutations. If you rearrange the digits, the numbers will be entirely different. This is because each unique arrangement is significant.

Mathematically, permutations of a set of items can be calculated using the formula:
  1. For selecting all items: \[ n! \] n! (n factorial) - where n is the total number of items.
  2. For selecting r items from n items:
    \[ P(n, r) = \frac{n!}{(n-r)!} \] \br \br This formula adjusts for cases where we're picking just a few items out of the whole set.

Always remember, if the order you arrange your items in affects the outcome, then you are dealing with permutations.
Combinations
Combinations are about selection without worrying about order. If we choose objects where the arrangement does not matter, we are talking about combinations. For example, in poker, a hand of five cards is considered the same no matter how you order those five cards.

The formula for combinations, where the order is ignored, is:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Here, we're choosing r items from a total of n items, and since order does not matter, we divide by the number of ways to arrange r items (which is r!).

Remember, if you simply choose items and their sequence doesn't affect the result, then it's a combination.
Order Matters
To decide whether a situation is a permutation or a combination, ask yourself if the order matters.

  • Order Matters: If changing the order results in a different outcome, then the order does matter, and you're looking at permutations.
  • Order Doesn't Matter: If changing the order does not affect the outcome, then the order doesn't matter, and you're looking at combinations.
Take a padlock as an example. Despite being called a 'combination,' it's actually a permutation since the order of numbers matters. Whereas in a lottery where you pick six numbers, the order of those numbers doesn't matter.

This is the key question to unlock whether you need permutations or combinations in your problem.
Selection of Objects
When we talk about selecting objects, we usually face two main scenarios: selecting with order consideration and selecting without considering the order.

  • Permutations: Selecting with order consideration. Example: license plate characters.
  • Combinations: Selecting without order consideration. Example: committee of politicians.
Think about a committee selection. Whether John is selected before Mary or Mary before John, it is still the same committee. However, if you were arranging people in seats where each person's position specifically matters, it would be a permutation. The subtle difference lies in whether the place or arrangement has significance.

Always check if the arrangement's significance in your problem is to determine if it's about permutations or combinations.
Probability in Mathematics
Probability involves predicting the likelihood of different outcomes. When dealing with permutations and combinations, probability can change based on whether permutations or combinations are being used.

  • Permutations: Where order affects probabilities. Example: specific phone numbers.
  • Combinations: Where order does not influence probabilities. Example: winning lottery numbers (if order does not matter).


Understanding whether you're working with a permutation or a combination is crucial for calculating the correct probability. For instance, the probability of selecting a winning lottery combination versus getting a correct permutation of digits in a game can be vastly different.

By understanding the concepts of permutations and combinations properly, you can apply them to calculate the probabilities of different scenarios accurately.

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Most popular questions from this chapter

Certain medical conditions are treated with a fixed dose of a drug administered at regular intervals. Suppose a person is given 2 mg of a drug each day and that during each 24 -hr period, the body utilizes \(40 \%\) of the amount of drug that was present at the beginning of the period. (a) Show that the amount of the drug present in the body at the end of \(n\) days is $$\sum_{i=1}^{n} 2(0.6)^{i}$$ (b) What will be the approximate quantity of the drug in the body at the end of each day after the treatment has been administered for a long period of time?

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{4}\left(3 i^{3}+2 i-4\right)$$

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Velma specializes in making different vegetable soups with carrots, celery, beans, peas, mushrooms, and potatoes. How many different soups can she make with any 4 ingredients?

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