Math Problem Statement
assignment comp.pdf
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solve it more clearly
Solution
Let's solve the first problem clearly.
Problem 1:
A committee of 6 is to be chosen from 10 men and 7 women such that it contains at least 3 men and 2 women. However, two particular women refuse to serve on the same committee. How many different ways can this be done?
Step-by-step solution:
Step 1: Understanding the conditions
- The committee consists of 6 people.
- The committee must contain at least 3 men and 2 women.
- There are 10 men and 7 women available.
- Two particular women cannot serve on the same committee.
Step 2: Identify possible gender combinations for the committee
Since the committee must contain at least 3 men and 2 women, the valid combinations of men and women are:
- 3 men and 3 women.
- 4 men and 2 women.
- 5 men and 1 woman.
Step 3: Calculate the number of ways to choose men and women for each combination
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For 3 men and 3 women:
- Number of ways to choose 3 men from 10 men = .
- Number of ways to choose 3 women from 7 women = .
However, two women refuse to serve together. Let's denote them as and . We need to subtract the cases where both and are chosen together.
- Number of ways to choose 3 women without restriction = .
- Number of ways to choose 3 women where both and are included: We must select 1 more woman from the remaining 5 (since and are already chosen), which is .
Hence, the total number of valid ways to choose 3 women is:
-
For 4 men and 2 women:
- Number of ways to choose 4 men from 10 men = .
- Number of ways to choose 2 women from 7 women = .
We need to exclude the cases where and are both chosen.
- Number of ways to choose 2 women without restriction = .
- Number of ways to choose 2 women where and are both chosen = 1 (since they are the only 2 women).
Hence, the total number of valid ways to choose 2 women is:
-
For 5 men and 1 woman:
- Number of ways to choose 5 men from 10 men = .
- Number of ways to choose 1 woman from 7 women = .
In this case, there are no restrictions on the woman chosen because only 1 woman is selected, so we don't have to worry about and both being chosen.
Step 4: Calculate the total number of ways for each scenario
-
3 men and 3 women:
-
4 men and 2 women:
-
5 men and 1 woman:
Step 5: Add the results from all the scenarios
The total number of ways to form the committee is the sum of the valid choices from all three cases.
If you'd like, I can calculate the numerical values for you or walk you through each combination calculation. Would you like more details?
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Probability
Set Theory
Formulas
nCr = n! / (r!(n-r)!)
Conditional combinations
Theorems
Combination theorem
Suitable Grade Level
Undergraduate level, Discrete Mathematics
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