Question 1190992
**a. Arrangements in a line:**

There are 10 people in total, so there are 10! (10 factorial) ways to arrange them in a line.

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

**b. Arrangements with men and women alternating around a table:**

Since there are 5 men and 5 women, they can alternate in two ways: M W M W M W M W M W or W M W M W M W M W M. 

First, arrange the men. There are 5 men, so there are 5! ways to arrange them. Similarly, there are 5! ways to arrange the women.

So, the total number of arrangements is 2 × 5! × 5! = 2 × 120 × 120 = 28,800.

**c. Arrangements in a line with men and women alternating:**

Similar to the table arrangement, there are two possible alternating patterns: M W M W M W M W M W or W M W M W M W M W M.

The number of ways to arrange the men is 5! and the number of ways to arrange the women is 5!.

So, the total number of arrangements is 2 × 5! × 5! = 2 × 120 × 120 = 28,800.

**Final Answers:**

a. 3,628,800 ways
b. 28,800 ways
c. 28,800 ways