Question 2 options:
A team of 9 people is to be formed from 7 men and 8 women.

(a) How many ways can the team be formed if the team must consist of 5 men and 4 women?

Now, the 9 people in the team (consisting of 5 men and 4 women) are to be seated in a row of 9 chairs.

How many ways can the 9 people be seated if

(b) they can sit anywhere?

(c) all the men must sit together?

To solve these problems, we need to use combinations and permutations. Let's break down each part:

### (a) Forming the team with 5 men and 4 women

To form a team of 5 men from 7 men and 4 women from 8 women, we use combinations. The combination formula is given by:

\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

- **Selecting 5 men from 7 men:**

  \[
  \binom{7}{5} = \frac{7!}{5! \times (7-5)!} = \frac{7 \times 6}{2 \times 1} = 21
  \]

- **Selecting 4 women from 8 women:**

  \[
  \binom{8}{4} = \frac{8!}{4! \times (8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
  \]

Thus, the total number of ways to form the team is:

\[
21 \times 70 = 1470
\]

### (b) Seating the 9 people in any order

Once the team is formed, we need to arrange 9 people in a row. This is a permutation problem, where we are arranging 9 distinct people:

- **Number of ways to seat 9 people:**

  \[
  9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880
  \]

### (c) Seating with all men together

If all the men must sit together, we can consider the group of 5 men as a single "block" or "unit." This reduces the problem to arranging 5 women and 1 block of men, totaling 6 units.

- **Arrange the 6 units:**

  \[
  6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
  \]

- **Arrange the 5 men within the block:**

  \[
  5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
  \]

Therefore, the total number of ways to seat the 9 people with all men together is:

\[
6! \times 5! = 720 \times 120 = 86400
\]

### Summary

- **(a)** Number of ways to form the team: **1470**
- **(b)** Number of ways to seat 9 people: **362880**
- **(c)** Number of ways to seat with all men together: **86400**

Would you like further details or have any questions? Here are some additional questions you might want to consider:

1. How do combinations differ from permutations?
2. Can the arrangement method be applied to circular seating?
3. What if no two women can sit together? How does that change the solution?
4. How would the solution change if one man or woman must be included in the team?
5. How can we verify the correctness of combination calculations using a calculator?

**Tip:** Remember that permutations are used when the order matters, and combinations are used when the order does not matter.

Explore how to form a team of 9 people with specific gender constraints and arrange them in various seating scenarios. Learn about combinations, permutations, and their applications in team formation and seating arrangements.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation