Possible Arrangements for Mixed Doubles Tennis Game

Given:
- Total players: 10 (6 men and 4 women)

To determine the number of ways in which the mixed doubles tennis game can be arranged, we need to consider the possible combinations of pairs that can be formed from the group of players.

1. Selecting the Men:
Since there are 6 men in the group, we need to choose 2 men to form a pair. This can be done using the combination formula.
Number of ways to select 2 men from 6 = C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 15

2. Selecting the Women:
Since there are 4 women in the group, we need to choose 2 women to form a pair. Again, we can use the combination formula.
Number of ways to select 2 women from 4 = C(4, 2) = 4! / (2! * (4-2)!) = 4! / (2! * 2!) = (4 * 3) / (2 * 1) = 6

3. Forming Mixed Doubles Pairs:
Now, we have selected 2 men and 2 women. We can form pairs by pairing each man with each woman. As there are 2 men and 2 women, there will be 2 * 2 = 4 possible pairs.

4. Arranging the Pairs:
Finally, the pairs can be arranged in the tennis game in any order. Since there are 4 pairs, the number of possible arrangements is equal to 4!.

Total number of ways to arrange the mixed doubles tennis game = Number of ways to select men * Number of ways to select women * Number of pairs * Number of arrangements
= 15 * 6 * 4 * 4!
= 15 * 6 * 4 * 4 * 3 * 2 * 1
= 180

Therefore, there are 180 ways in which the mixed doubles tennis game can be arranged from a group of 10 players consisting of 6 men and 4 women.