Final answer:
To find the number of ways to pair off 10 boys and 10 girls, one calculates the factorial of 10, which equals 3,628,800. This represents the number of distinct pairings the matchmaker can create.
Step-by-step explanation:
The question at hand involves determining the number of ways the village matchmaker can pair off 10 boys and 10 girls into marriages. This is a problem of combinatorics, specifically calculating permutations.
Since there are 10 girls and each girl can be paired with any of the 10 boys, we consider the first girl having 10 choices. After she is paired, the next girl will have 9 choices, and this pattern continues with the number of choices decreasing by 1 for each subsequent girl as the boys are paired off.
To find the total number of pairings, we would calculate the factorial of the number of girls or boys (since there are an equal number of each). This method of calculation is often represented as '10!' (factorial of 10), which is the product of all positive integers up to 10:
10! = 10 × 9 × 8 × ... × 1.
The result of 10! will give us the number of ways the matchmaker can pair off the children. The factorial of 10 is 3,628,800, so that is the number of different ways the matchmaker can arrange the marriages.