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How many different ways are there for n families to line up single file for an amusement park ride, if each family consists of two parents and k children, and if the members of each family must all stand together, with the children between the parents?

User Rizwana
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Final answer:

The number of ways n families can line up for an amusement park ride, with each family standing together and parents at the ends, is calculated as n! (n factorial). For 4 families, there are 4! = 24 different lineups.

Step-by-step explanation:

The student is asking about the number of different ways n families can line up single file for an amusement park ride, assuming that within each family, two parents must stand at either end with their k children between them. Since the families must stay together, we can consider each family as a single unit when determining the lineup order. The number of ways to arrange n units is n! (n factorial), which represents the product of all positive integers from 1 up to n.

In this specific case, if there are 4 families (n=4), the number of different ways they can line up is 4!, which equals 4 × 3 × 2 × 1, resulting in 24 different lineups. However, if each family has a different number of children (k), there could be additional permutations within each family based on the order of the children between the parents, but since the question does not provide k or ask for these permutations, we only consider the order of the families as a whole.

User Kwong
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