The problem asks to identify the incorrect statement based on the seating arrangement of five persons: A, B, C, D, and E.
Analyzing Seating Constraints
- There are 5 positions in the row: 1, 2, 3, 4, 5.
- C is in the middle: C is at position 3. Arrangement: $_ _ C _ _$
- D is at an extreme end: D is either at position 1 or position 5.
- At least two persons between B and E: The number of positions between B and E must be 2 or more. This implies the difference in their positions must be at least 3 (e.g., position 1 and 4, or 1 and 5, or 2 and 5). Mathematically, $|Pos(B) - Pos(E)| \ge 3$.
Determining Possible Arrangements
We consider the two cases for D's position:
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Case 1: D is at position 1.
Arrangement: $D _ C _ _$
Positions available for B and E are 2, 4, 5. To satisfy $|Pos(B) - Pos(E)| \ge 3$, B and E must occupy positions 2 and 5. The remaining person, A, takes position 4.
- Subcase 1a: B at 2, E at 5. Arrangement: $D B C A E$. (Positions 3 and 4 are between B and E).
- Subcase 1b: E at 2, B at 5. Arrangement: $D E C A B$. (Positions 3 and 4 are between E and B).
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Case 2: D is at position 5.
Arrangement: $_ _ C _ D$
Positions available for B and E are 1, 2, 4. To satisfy $|Pos(B) - Pos(E)| \ge 3$, B and E must occupy positions 1 and 4. The remaining person, A, takes position 2.
- Subcase 2a: B at 1, E at 4. Arrangement: $B A C E D$. (Positions 2 and 3 are between B and E).
- Subcase 2b: E at 1, B at 4. Arrangement: $E A C B D$. (Positions 2 and 3 are between E and B).
The possible valid arrangements are: $D B C A E$, $D E C A B$, $B A C E D$, $E A C B D$.
Evaluating the Statements
Now, let's check each statement against these arrangements:
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Statement 1: E can be on extreme left.
This is true in arrangement $E A C B D$, where E is at position 1.
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Statement 2: E can be on extreme right.
This is true in arrangement $D B C A E$, where E is at position 5.
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Statement 3: A cannot be on extreme left.
In all derived arrangements (A is at position 4 or 2), A is never at the extreme left (position 1). This statement is true.
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Statement 4: A is always a neighbour of B or D.
- In $D B C A E$: A is at position 4. Neighbours are C (3) and E (5). A is not adjacent to B (2) or D (1). This contradicts the statement.
- In $D E C A B$: A is at position 4. Neighbours are C (3) and B (5). A is adjacent to B. Statement holds.
- In $B A C E D$: A is at position 2. Neighbours are B (1) and C (3). A is adjacent to B. Statement holds.
- In $E A C B D$: A is at position 2. Neighbours are E (1) and C (3). A is not adjacent to B (4) or D (5). This contradicts the statement.
Since the statement "A is *always* a neighbour of B or D" is false for arrangements $D B C A E$ and $E A C B D$, the statement itself is incorrect.
Conclusion
The statement that is incorrect is "A is always a neighbour of B or D".
Important Questions from Seating Arrangement (Notes)
- Five friends are sitting on a bench in the following order :
(i) P is sitting next to Q and R is next to S.
(ii) S is not sitting with T; T is on the extreme left hand side of the bench and R is on second position from the right hand side.
(iii) P is on the right hand side of Q and to the right side of T. (iv) P and R sitting together
Who is sitting exactly in the middle ? Seven people, C, D, E, F, L, M and N, are sitting in a row, facing north.
Only two people sit to the right of C. Only two people sit between C and L. Only two people sit between E and N. N sits to the immediate left of C. M sits to the immediate right of F. Who sits at the third position from the left end of the row?- Seven people, A, B, C, D, E, F and G, are sitting in a row, facing north. Only three people sit between E and D. A sits to the immediate left of D. No one sits to the right of B. Only two people sit between B and A. G sits to the immediate right of F. How many people sit to the right of F?
Seven boxes, A, B, C, D, E, F and G, are kept one over the other but not necessarily in the same order. Only A is kept above F. Only D is kept between F and C. Only E is kept below B. How many boxes are kept below C?
P, Q, R, S, T, U and V are sitting in a row, facing north. No one sits to the left of U. Only four people sit between U and R. Only three people sit to the right of V. P sits to the immediate left of S. Q is not an immediate neighbour of V. How many people sit between P and T?
