a/A class test consists of 4 Algebra questions,W ,X,Y and Z, and 3 Geometry questions,A,B,C and D.The teacher decides that the questions should be arranged in two sections.Algebra followed by Geometry,with questions in each section arranged in a random order Find the number of arrangement in which questions W and B are next to each other. Find the number of arrangements in which questions X and D are separated by more than four other subjects

W and B are next to each other:

let's put the there, then we have
? ? ? W B ? ? ?
3*2*1 W B 3*2*1
number of ways = 3*2*1*1*1*3*2*1 = 36

number of arrangements in which questions X and D are separated by more than four other subjects:
We already know that there are 36 cases where a X and D would be in specific places, (see first part, only the names would change)
case 1 , X is in the 1st position, so we could have
X ? ? ?|? ? D ? -----> 36 of those
X ? ? ?|? ? ? D -----> 36 of those
case 2:, the X is in the 2nd position
? X ? ?|? ? ? D -----> 36 of those
there are no other possibilities, so you
have 3(36) or 108 such cases.

Three postings of the same question where you say "there are 3 geometry questions , A, B, C, and D." That's four!

Clarify before I answer it

You are right.It was a typo

So what is the correct version ????

A class test consists of 4 Algebra questions,W ,X,Y and Z, and 4 Geometry questions,A,B,C and D.The teacher decides that the questions should be arranged in two sections.Algebra followed by Geometry,with questions in each section arranged in a random order Find the number of arrangement in which questions W and B are next to each other. Find the number of arrangements in which questions X and D are separated by more than four other subjects.

To find the number of arrangements in which questions W and B are next to each other, we can treat W and B as a single entity. This means that we have 8 entities (W/B, X, Y, Z, A, C, D) to arrange in two sections (Algebra and Geometry). The W/B entity can be placed in two different positions within this group (either at the beginning or at the end). Once the W/B entity is placed, the remaining 7 entities can be arranged within their respective sections in any order.

Therefore, the number of arrangements in which questions W and B are next to each other is 2 * (7!) = 10,080.

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 different possible positions for X and D within the 8 entities (W, X, Y, Z, A, B, C, D).

There are three possible cases:
1. X is placed before D.
2. X is placed after D.
3. X and D are separated by exactly one other subject.

For case 1, we have the following possible positions: (X, D, ...), (Y, X, D, ...), (Z, Y, X, D, ...), (A, Z, Y, X, D, ...), (B, A, Z, Y, X, D, ...), (C, B, A, Z, Y, X, D, ...), (D, C, B, A, Z, Y, X, ...). In this case, there are 7 entities remaining that can be arranged in their respective sections in any order.

For case 2, we have the following possible positions: (D, X, ...), (D, Y, X, ...), (D, Z, Y, X, ...), (D, A, Z, Y, X, ...), (D, B, A, Z, Y, X, ...), (D, C, B, A, Z, Y, X, ...), (C, D, B, A, Z, Y, X, ...). Again, there are 7 entities remaining that can be arranged in any order.

For case 3, X can be placed in one of the 4 positions, and D can be placed in one of the 3 remaining positions. The remaining 6 entities can be arranged in any order.

Therefore, the total number of arrangements in which questions X and D are separated by more than four other subjects is: (7! * 7) + (7! * 7) + (4 * 3 * 6!) = 12,096.