In how many ways can 15 stduents be seated in a row such that the 2 most talkative children never sit together?
answer choose are
a. 14!,14!
b. 15,14!
c.14!
d. 14!13
e. 15!
To solve this problem, we can consider two scenarios:
1. The most talkative children are sitting at the ends.
2. The most talkative children are sitting somewhere in the middle.
Scenario 1: The most talkative children are sitting at the ends.
In this case, we can treat the two most talkative children as a single entity. So effectively, we have 14 entities (13 students + 1 entity composed of the most talkative children) to arrange in a row. The number of ways to arrange these entities is given by 14!.
Scenario 2: The most talkative children are sitting somewhere in the middle.
In this case, we can place the most talkative children in any of the 14 gaps between the other students. Once the most talkative children are placed, we can arrange the remaining 13 students in the remaining 13 gaps. The number of ways to arrange the remaining students is given by 13!.
Considering both scenarios, the total number of ways to arrange the students is the sum of the number of ways in each scenario, which is 14! + 13!.
Therefore, the answer is not listed among the available options.