I'm tutoring someone 1st year arts math and they have been given this difficult question that I cannot mind boggel. It goes as follows: There are 6 couples at a party. Each can shake another persons hand once, but they can never shake their spouses hand. At the end of the night when a husband asks his wife and all the other people how many hands they shook he gets eleven different answers. Then the question asks how many hands did the husband's wife shake? I'm very confused by it. If anyone could help that would be great.
With 6 couples, there are 12 people. Considering that you don't want to shake hands with yourself or your spouse, you — or your spouse — could shake hands with 10 other people. The word, "could," implies potential, not actual activity. It could be that many of the others did not shake hands with all the potential persons.
Thus someone shakes 0 hands, someone 1, someone 2,...to someone 10, giving 11 different answers. However, I don't know how to determine which one is the wife.
I hope this helps. Thanks for asking.
To solve this problem, let's break it down step by step.
Step 1: Understanding the problem
We have 6 couples at a party. Each person shakes hands with others, excluding their spouse. At the end of the night, when a husband asks his wife and everyone else how many hands they shook, he receives eleven different answers. We want to determine how many hands the husband's wife shook.
Step 2: Identify the possibilities for handshakes
Let's consider the possibilities for the number of handshakes each person can have. Since each person shakes hands with everyone except their spouse, the number of handshakes for each person can be any number from 0 to 10.
Step 3: Analyzing the answers
When the husband asks everyone how many hands they shook, he receives eleven different answers. Since there are only 6 couples, this means there must be some repetition in the answers. We need to find a pattern to understand this repetition.
Step 4: Finding the pattern
Since there are 6 couples and each person shakes hands with everyone except their spouse, the maximum number of different answers the husband can receive is 12 (if each person shakes hands with exactly one person of each couple). However, we are told that the husband receives only 11 different answers. This means one of the couples must have given the same answer as another couple.
Step 5: Identifying the couple with the same answer
To determine which couple gave the same answer, let's consider the number of handshakes for each possible answer. We can create a table to list the number of handshakes and the frequency at which each answer appears.
Number of Handshakes | Frequency
----------------------------------
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
----------------------------------
Step 6: Analyzing the frequency table
Looking at the table, we know there are 11 different answers given. However, we need to find the couple that has the same answer as another couple. We notice that 12 is missing from the frequency table. This suggests that one couple gave the answer of 12, which caused the repetition in the responses.
Since each person shakes hands with everyone except their spouse, the maximum number of handshakes for any individual is 10. Therefore, the couple who gave the answer of 12 must be the husband and wife in question. This implies that the husband's wife shook hands with 12 people.
Step 7: Conclusion
The wife of the husband in question shook hands with 12 people at the party.
To recap, by understanding the given information, analyzing the possibilities of handshakes, and identifying the repetition in the answers, we determined that the wife of the husband in question shook hands with 12 people.