There Are 5 Women 4 Men In A Club . A Term Of 4 Has Be Chosen . How Many Different Term Can Be Chosen If There Must Be Either Exactly Two Women In The Term

Introduction

In a club consisting of 5 women and 4 men, a term of 4 members needs to be chosen. The condition is that the term must have exactly two women. In this scenario, we need to determine the number of different terms that can be chosen under this condition.

Choosing Exactly Two Women

To start, we need to choose exactly two women from the 5 available women in the club. This can be calculated using the combination formula, denoted as C(n, r) = n! / (r!(n-r)!), where n is the total number of items, and r is the number of items to be chosen.

For choosing 2 women from 5, the calculation is as follows:

C(5, 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / ((2 × 1)(3 × 2 × 1)) = (120) / ((2)(6)) = 120 / 12 = 10

So, there are 10 different ways to choose exactly two women from the 5 women in the club.

Choosing the Remaining Two Members

After choosing the two women, we need to choose the remaining two members from the 4 men in the club. This can also be calculated using the combination formula.

For choosing 2 men from 4, the calculation is as follows:

C(4, 2) = 4! / (2!(4-2)!) = 4! / (2!2!) = (4 × 3 × 2 × 1) / ((2 × 1)(2 × 1)) = (24) / ((2)(2)) = 24 / 4 = 6

So, there are 6 different ways to choose the remaining two members from the 4 men in the club.

Calculating the Total Number of Terms

To find the total number of different terms that can be chosen with exactly two women, we need to multiply the number of ways to choose the two women by the number of ways to choose the remaining two members.

Total number of terms = Number of ways to choose 2 women × Number of ways to choose 2 men = 10 × 6 = 60

Therefore, there are 60 different terms that can be chosen from the club with exactly two women.

Conclusion

In conclusion, when choosing a term of 4 members from a club consisting of 5 women and 4 men, with the condition that the term must have exactly two women, there are 60 different terms that can be chosen. This is calculated by multiplying the number of ways to choose the two women by the number of ways to choose the remaining two members.

Additional Considerations

It's worth noting that this calculation assumes that the order of selection does not matter, and that the same term cannot be chosen multiple times. If the order of selection does matter, or if the same term can be chosen multiple times, the calculation would need to be adjusted accordingly.

Example Use Cases

This problem can be applied to various real-world scenarios, such as:

• Choosing a team of 4 members from a group of 5 women and 4 men, with the condition that the team must have exactly two women.
• Selecting a committee of 4 members from a group of 5 women and 4 men, with the condition that the committee must have exactly two women.
• Determining the number of different ways to choose a group of 4 members from a club consisting of 5 women and 4 men, with the condition that the group must have exactly two women.

Final Thoughts

In conclusion, the problem of choosing a term of 4 members from a club consisting of 5 women and 4 men, with the condition that the term must have exactly two women, is a classic example of a combinatorics problem. By using the combination formula, we can calculate the total number of different terms that can be chosen under this condition.