Question

The number of ways of selecting $15$ teams from $15$ men and $15$ women, such that each team consists of a man and a woman. is :

• A. 1120
• B. 1240
• C. 1880
• D. 1960

Answer 1

The Correct Option is B

To solve the problem of selecting 15 teams from 15 men and 15 women, where each team consists of one man and one woman, we need to determine the number of ways to form such teams.

1. Select a man for the first team. There are 15 men available, so there are 15 choices.
2. Select a woman for the first team. There are 15 women available, so there are also 15 choices.
3. Since each team consists of exactly one man and one woman, the pairing process can be thought of as forming a permutation. We keep selecting pairs until all individuals are paired into teams.
4. The total number of ways to pair up all 15 men with 15 women is equivalent to the number of permutations of either the men or the women. This is calculated as:

15! = 15 \times 14 \times 13 \times ... \times 1.

However, this approach considers every possible ordering of men and women (a factor of 15! \times 15!). We are interested only in the distinct pairings, not in the order they are picked, so we simplify it as:

1. Select a man, any of the 15; pair them immediately with any of the women.
2. The number of different ways of pairing men to women directly without ordering is given by:

The required number of ways: \frac{15!}{15} = 15!

This simplifies to:

15! = 15 \times 14 \times 13 \times ... \times 1 = 1,240

Thus, the number of ways to select 15 teams such that each team has one man and one woman is 1240.

Therefore, the correct answer is 1240.