To find the number of arrangements in which questions W and B are next to each other, we can treat them as a single block. This means that instead of having 4 Algebra questions and 3 Geometry questions, we now have 3 "blocks" - the block with questions W and B as one, and the other two blocks remaining as Algebra and Geometry questions.
Let's find the number of arrangements for each block and multiply them together to get the final result:
1. The number of arrangements for the block with questions W and B:
Since W and B need to be next to each other, we can treat them as a single entity. This block has 2 questions, and we can arrange them in 2! = 2 factorial ways.
2. The number of arrangements for the Algebra block (excluding W and B):
We have 4 - 2 = 2 remaining Algebra questions (X and Y) in this block. We can arrange them in 2! ways.
3. The number of arrangements for the Geometry block:
We have 3 Geometry questions (A, C, and D) in this block. We can arrange them in 3! ways.
To find the total number of arrangements:
Total arrangements = (Number of arrangements for the block with W and B) * (Number of arrangements for the Algebra block) * (Number of arrangements for the Geometry block)
Total arrangements = 2! * 2! * 3!
Now let's calculate this:
Total arrangements = 2 * 2 * 3 * 2 * 1 = 24
So, there are 24 arrangements in which questions W and B are next to each other.
Now let's move on to the second part of the question.
To find the number of arrangements in which questions X and D are separated by more than four other subjects, we need to consider the possible number of subjects separating them.
If there are 5 or more subjects between X and D, we can arrange them in various ways.
1. Let's first consider 5 subjects between X and D:
There are 5 Algebra questions (W, X, Y, Z, and one more) and 3 Geometry questions (A, B, and C) that can be placed between X and D.
The number of ways to arrange these 8 questions is 8!.
However, X and D can swap positions, so we need to multiply by 2 to account for this possibility.
The number of arrangements with 5 subjects between X and D = 8! * 2.
2. Similarly, we can consider 6, 7, and 8 subjects between X and D:
The number of arrangements with 6 subjects between X and D = 9! * 2.
The number of arrangements with 7 subjects between X and D = 10! * 2.
The number of arrangements with 8 subjects between X and D = 11! * 2.
To find the total number of arrangements:
Total arrangements = (Number of arrangements with 5 subjects) + (Number of arrangements with 6 subjects) + (Number of arrangements with 7 subjects) + (Number of arrangements with 8 subjects)
Total arrangements = (8! * 2) + (9! * 2) + (10! * 2) + (11! * 2)
Now let's calculate this:
Total arrangements = 2 * (8! + 9! + 10! + 11!)
Using a calculator or software, you can find the numerical value of this expression.
So, the total number of arrangements in which questions X and D are separated by more than four other subjects depends on the calculations explained above.