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.
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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 and arrange the rest of the questions (X, Y, Z, A, C, and D) together with this entity.
1. Treating W and B as a single entity, we have 7 entities to arrange: WB, X, Y, Z, A, C, and D.
2. Since these entities can be arranged in any order, the number of arrangements would be the factorial of the total number of entities.
Therefore, the number of arrangements in which questions W and B are next to each other would be 7!.
To find the number of arrangements in which questions X and D are separated by more than four other subjects, we can consider two cases:
Case 1: X is followed by D.
In this case, we can treat XD as a single entity and arrange the rest of the questions (W, Y, Z, A, B, and C) together with this entity.
1. Treating XD as a single entity, we have 7 entities to arrange: XD, W, Y, Z, A, B, and C.
2. Since these entities can be arranged in any order, the number of arrangements would be the factorial of the total number of entities.
Therefore, the number of arrangements in which X is followed by D would be 7!.
Case 2: D is followed by X.
In this case, we can treat DX as a single entity and arrange the rest of the questions (W, Y, Z, A, B, and C) together with this entity.
1. Treating DX as a single entity, we have 7 entities to arrange: DX, W, Y, Z, A, B, and C.
2. Since these entities can be arranged in any order, the number of arrangements would be the factorial of the total number of entities.
Therefore, the number of arrangements in which D is followed by X would be 7!.
To find the total number of arrangements in which questions X and D are separated by more than four other subjects, we can add the number of arrangements in Case 1 and Case 2.
Total number of arrangements = Number of arrangements in Case 1 + Number of arrangements in Case 2
Total number of arrangements = 7! + 7!