In how many ways can 4 men and 3 women arrange themselves in a row for picture taking if the men and women must stand in alternate positions? 

In how many ways can 4 men and 3 women arrange themselves in a row for picture taking if the men and women must stand in alternate positions?

To solve this problem, we can treat the arrangement of men and women separately. Since they must alternate, we can first arrange the men in all possible ways and then arrange the women in all possible ways, and then multiply the two results together.

Given:

• 4 men (M1, M2, M3, M4)
• 3 women (W1, W2, W3)

For the men, they can be arranged in 4! (4 factorial) ways (4 men * 3 men * 2 men * 1 men), since there are 4 positions and 4 choices for the first position, 3 choices for the second position, 2 choices for the third position, and only 1 choice for the last position.

For the women, they can be arranged in 3! (3 factorial) ways (3 women * 2 women * 1 women), following the same logic.

So, the total number of ways to arrange the men and women alternately is:

4! * 3! = 24 * 6 = 144 ways.

Therefore, there are 144 ways in which 4 men and 3 women can arrange themselves in a row for picture-taking if they must stand in alternate positions.

What is Combinatorics?

Combinatorics is a branch of mathematics concerned with counting, arranging, and analyzing finite sets of objects according to certain principles or rules. It deals with problems of selection, arrangement, and combination of elements from a given set, often in contexts such as permutations, combinations, and partitions.

Combinatorics has applications in various fields including computer science, cryptography, probability theory, and optimization. It provides tools and techniques for solving problems involving discrete structures and finite systems, making it a fundamental area of study in mathematics with broad interdisciplinary applications.