To solve this problem, let's break it down step by step.
We are asked to select a team of four students from a total of 12 students, with the following conditions:
1. Two particular students refuse to be together.
2. Other two particular students wish to be together only.
### Step 1: Treat the two students who wish to be together as one unit.
Since the two students who wish to be together only must always be together, we can treat them as a single unit or "block." Therefore, instead of selecting 4 students, we now need to select 3 units:
- One "block" made up of the two students who wish to be together.
- Two additional students from the remaining 10 students (since we already considered the "block").
### Step 2: Select the block and two other students.
The number of ways to choose the "block" of two students is just 1, as these two students must always be together.
Next, we select two students from the remaining 10 students, excluding the two students who refuse to be together. The number of ways to do this is given by the combination formula \( C(n, r) \), which calculates the number of ways to choose \( r \) objects from \( n \) objects:
\[
C(10, 2) = \frac{10 \times 9}{2 \times 1} = 45
\]
### Step 3: Subtract the cases where the two students who refuse to be together are both chosen.
We must now exclude the cases where the two students who refuse to be together are selected in the team, as they are not allowed to be together. If both these students are selected, we have:
- One "block" of two students (who must be together).
- The two students who refuse to be together, selected from the remaining students.
Since these two students must not be together, we subtract the number of ways in which both students who refuse to be together are selected. The number of ways to select the "block" and the two students who refuse to be together is 1 way for the "block," and 0 ways for selecting the students who refuse to be together, as we already know they cannot be together.
### Step 4: Final answer
The total number of valid selections is calculated as follows:
- Number of ways to select a valid team = Total number of ways to select the 4 students − Invalid selections where the two students who refuse to be together are selected.
We are looking for the option that matches the final result of the correct number of selections. Hence, we
