Final answer:
To determine the number of ways 7 people can be arranged in line with Kara and Florence sitting next to each other, we treat the pair as one entity, thus arranging 6 entities (720 ways), and then multiply by 2 for the internal arrangement of the pair, resulting in 1440 different arrangements.
Step-by-step explanation:
The problem involves determining the number of ways 7 people can be arranged in a line at the movies such that two specific people, Kara and Florence, must be sitting next to each other. To solve this, we first consider Kara and Florence as a single entity since they must be together. Therefore, instead of arranging 7 individuals, we are arranging 6 entities (5 individuals + 1 pair).
Firstly, we find the number of ways to arrange these 6 entities, which is 6!, as each entity can be in any of the 6 positions. The calculation is as follows: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
Secondly, within this entity, Kara and Florence can be arranged in 2! ways since it does not matter if Kara is first or Florence is first. The calculation for their internal arrangement is 2! = 2 × 1 = 2.
Finally, to get the total number of arrangements, we multiply the two results: 720 (for the 6 entities) × 2 (for the internal arrangement of Kara and Florence) = 1440.
Therefore, there are 1440 different ways the people can be arranged in line with the condition that Kara and Florence must be sitting next to each other.