91Ó°ÊÓ

Thousands of jokes have been told about marriage and divorce. Exercises \(61-68\) are based on the following observations: \- "By all means, marry; if you get a good wife, you'll be happy. If you get a bad one, you'll become a philosopher:"Socrates \- "My wife and I were happy for 20 years. Then we met." \- Rodney Dangerfield \- "Whatever you may look like, marry a man your own age As your beauty fades, so will his eyesight." - Phyllis Diller \- "Why do Jewish divorces cost so much? Because they're worth it. "Henny Youngman \- I think men who have a pierced ear are better prepared for marriage. They've experienced pain and bought jewelry." \- Rita Rudner \- "For a while we pondered whether to take a vacation or get a divorce. We decided that a trip to Bermuda is over in two weeks, but a divorce is something you always have."-Woody Allen In how many ways can people select their two favorite jokes from these thoughts about marriage and divorce?

Step by step solution

01

Identify the total number of items

Identify the total number of jokes from which selection will be made. Based on the details provided in the exercise, there are a total of 7 jokes.

02

Identify how many items are to be selected

Identify the number of jokes to be selected as favorites. The exercise specifies that 2 jokes are to be chosen as favorites.

03

Apply the combination formula

The number of ways to select 2 jokes out of 7 is determined by the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\). By substituting `n=7` (total jokes) and `k=2` (jokes to select) into the formula, compute the total number of ways to choose 2 favorite jokes out of 7.

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Key Concepts

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

Understanding the Combination Formula

In combinatorics, when we need to choose items from a larger set where the order of selection does not matter, we use the combination formula. This is crucial for solving problems like the one in the exercise, where we want to select jokes.
The combination formula is mathematically represented as:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

Here:

• \( n \) is the total number of items (7 jokes in our exercise).
• \( k \) is the number of items to choose (2 jokes to be selected).
• \( n! \) denotes the factorial of \( n \), which means multiplying all whole numbers from \( n \) down to 1.

The combination formula helps us determine the number of different ways to choose \( k \) items from \( n \) items without regard to order. In our case, substituting the numbers gives us \( C(7, 2) = \frac{7!}{2!(7-2)!} \). Calculating this reveals there are 21 different combinations of choosing 2 jokes from 7.

Permutations and Combinations: Key Differences

Combinatorics includes permutations and combinations, both of these are methods for counting arrangements of items. Understanding the difference between them is vital.

• Permutations: Concerned with the arrangement of items where order matters. For example, choosing 2 jokes and arranging them differ from just choosing them.
• Combinations: Focus on selection where the order does not matter. This is what the exercise called for. Simply picking two jokes without concern for which is picked first.

If we were dealing with permutations, we’d use a different formula: \( P(n, k) = \frac{n!}{(n-k)!} \).
This formula accounts for all possible orderings of selections. Since only combinations were needed in the exercise, our focus remains on the combination formula, where arrangements within a selected group do not change the outcome.

Applying Mathematical Reasoning

Mathematical reasoning is a process of following logic to reach a valid conclusion. It’s a critical skill in solving combinatorics problems.
For the exercise, the reasoning starts by distinctly identifying the elements involved: the total jokes (\( n = 7 \)) and the number of choices (\( k = 2 \)).

• First, you recognize the need to apply a concept—not all problems demand the same approach. The key is understanding that we need combinations, not permutations.
• Next, methodically apply the appropriate formula, substituting known values, \( n \) and \( k \).
• Finally, perform the operations correctly to arrive at the correct result, \( C(7, 2) = 21 \).

This logical approach—understanding the problem, applying the appropriate formula, and careful calculation—ensures successful problem-solving. This kind of reasoning helps link theoretical understanding to practical application, providing clarity and accuracy in mathematical tasks.