Question

There are 11 students enrolled in a history class. Student Aisha and student Kareem doesn’t get along well. The teacher needs to select 6 students from the classroom for a contest without putting Aisha and Kareem together to the team — Aisha or Kareem can be included in the team but just cannot be included together – How many ways are there to select the team?

Answer 1

To determine the number of ways to select a team of 6 students from a class of 11, while ensuring that Aisha and Kareem are not on the same team, we can use combinatorics.

First, let's consider the total number of ways to select 6 students from a class of 11 without any restrictions. This can be calculated using the combination formula:

C(n, r) = n! / (r!(n-r)!)

Where n is the total number of students (11 in this case) and r is the number of students to be selected (6 in this case).

Using this formula, we can calculate the total number of ways to select 6 students from a class of 11:

C(11, 6) = 11! / (6!(11-6)!) = 462

So there are 462 different ways to select a team of 6 students without any restrictions.

Now, let's consider the cases where Aisha and Kareem cannot be on the same team. We can break this down into two scenarios:

1. Aisha is selected for the team:

In this case, we need to select 5 more students from the remaining 10 (excluding Aisha and Kareem). Using the combination formula again:

C(10, 5) = 10! / (5!(10-5)!) = 252

So there are 252 different ways to select a team of 6 students with Aisha included but not Kareem.

2. Kareem is selected for the team:

Similarly, we need to select 5 more students from the remaining 10 (excluding Aisha and Kareem):

C(10, 5) = 10! / (5!(10-5)!) = 252

Again, there are 252 different ways to select a team of 6 students with Kareem included but not Aisha.

However, we need to consider that in both of these scenarios, there might be cases where both Aisha and Kareem are selected along with other students. To exclude these cases, we need to subtract the number of ways they can be selected together from the total.

To calculate this, we can treat Aisha and Kareem as a single entity and select 4 more students from the remaining 9 (excluding Aisha, Kareem, and the previously selected students):

C(9, 4) = 9! / (4!(9-4)!) = 126

Therefore, there are 126 different ways to select a team of 6 students with both Aisha and Kareem included.

To find the final number of ways to select a team of 6 students without putting Aisha and Kareem together, we subtract the cases where they are selected together from the total:

462 - 126 = 336

So there are 336 different ways to select a team of 6 students from a class of 11 while ensuring that Aisha and Kareem are not on the same team.

Step-by-step explanation: