A debate team consists of 5 freshmen and 4 sophomores. (Everyone is distinguishable.) In how many ways can they stand in line, so at least two of the sophomores are standing next to each other, and at least two of the freshmen are at the ends of the line?
To determine the number of ways the team can stand in line, satisfying the given conditions, we can use the principle of inclusion-exclusion.
First, let's find the total number of ways the team can arrange themselves without any restrictions. Since there are 9 students in total (5 freshmen and 4 sophomores), we can arrange them in 9! ways.
Next, let's consider the restriction that at least two freshmen must be at the ends of the line. There are 5 freshmen, so we have 5 options for the first position and 4 options for the last position. The remaining 7 students (4 sophomores and 3 freshmen) can be arranged among the 7 positions in 7! ways.
Now, let's consider the restriction that at least two sophomores must be standing next to each other. We will use the principle of inclusion-exclusion.
Let's assume the two sophomores standing next to each other as a single entity. This entity can be placed in 8 different positions: 7 positions between students and 1 position at the beginning or end of the line.
Inside this entity, the two sophomores can be arranged in 2! ways.
The remaining students (3 freshmen, 2 sophomores, and 2 freshmen) can be arranged among the remaining 7 positions in 7! ways.
Therefore, the total number of arrangements where at least two freshmen are at the ends of the line and at least two sophomores are standing next to each other can be calculated using inclusion-exclusion principle as follows:
Total arrangements = (Total arrangements) - (Arrangements with no freshmen at the ends) - (Arrangements with no sophomores standing next to each other) + (Arrangements with no freshmen at the ends and no sophomores standing next to each other)
Total arrangements = 9! - (5 * 7! * 2!) - (8 * 7!)
Now we can calculate this value:
Total arrangements = 362,880 - (5 * 5,040 * 2) - (8 * 5,040)
Total arrangements = 362,880 - 50,400 - 40,320
Total arrangements = 272,160
Therefore, there are 272,160 ways the debate team can stand in line, satisfying the given conditions.