Final Answer:
The number of possible arrangements for the couples taking a picture is
(5 factorial), which is equal to 120.
Step-by-step explanation:
In combinatorics, when dealing with permutations, the factorial represents the product of all positive integers up to . In this scenario, there are 5 pairs of couples, and each pair can be arranged among themselves in
ways (considering the wife and husband as a pair). Therefore, the total number of arrangements is for the wives in the first row.
Additionally, the husbands standing in the second row can also be arranged in ways, as each husband can take any of the 5 positions. Multiplying the number of arrangements for the wives by the number of arrangements for the husbands gives
Simplifying this expression provides the final answer of 120, which is the total number of distinct arrangements for the couples taking a picture. Each arrangement represents a unique configuration where each wife is paired with a husband, forming a diverse set of possibilities for the group photo.