Seven Cars P, Q, R, S, T, U and V are parked in a row not necessarily in that order. The cars T and U should be parked next to each other. The cars S and V also should be parked next to each other, whereas P and Q can’t be parked next to each other. Q and S must be parked next to each other. R is parked to the immediate right of V. T is parked to the left of U. Based on the above statements, the only incorrect option given below is
Correct Answer :
There are two cars parked in between Q and V.
Solution :
The correct option is: There are two cars parked in between Q and V.
Let us analyze the given conditions step-by-step to find the possible arrangements of the seven cars (P, Q, R, S, T, U, and V) parked in a row from left to right:
1. T and U are parked next to each other, and T is parked to the left of U: This means they form a block (T U).
2. S and V are parked next to each other: They form a block of (S, V) in some order.
3. Q and S must be parked next to each other: Combined with the previous condition, Q, S, and V must be adjacent. Thus, they must form either the block (Q S V) or (V S Q).
4. R is parked to the immediate right of V: This means R must be placed immediately after V, forming the block (V R). Let us see how this affects our combinations from step 3:
- If the block was (Q S V), placing R to the immediate right of V gives the block (Q S V R). This is a valid arrangement of 4 cars.
- If the block was (V S Q), we cannot place R to the immediate right of V because S is already to the immediate right of V. Thus, this case is impossible.
Therefore, the cars Q, S, V, and R must be parked in the exact order: (Q S V R).
5. P and Q cannot be parked next to each other: This means P cannot be placed immediately to the left of Q.
Now, let us place our blocks and the remaining car P in a row of 7 positions. The blocks we have are:
- Block 1: (T U) (occupies 2 consecutive positions)
- Block 2: (Q S V R) (occupies 4 consecutive positions)
- Individual car: P
Since Block 2 occupies 4 positions and Block 1 occupies 2 positions, together with P they occupy 7 positions. Let us consider the relative arrangement of Block 1, Block 2, and P:
Case A: Block 2 is to the left of Block 1.
The order could be:
- (Q S V R) P (T U): Here, P is between R and T. Since P is not next to Q, this is a valid arrangement.
- (Q S V R) (T U) P: Here, Block 1 is next to Block 2, and P is at the right end. This is a valid arrangement: Q S V R T U P.
Case B: Block 1 is to the left of Block 2.
The order could be:
- (T U) P (Q S V R): Here, P is between U and Q. But P cannot be next to Q, so this arrangement is invalid.
- P (T U) (Q S V R): Here, P is at the extreme left end, followed by T, U, and then Q, S, V, R. Since U is next to Q, and P is not next to Q, this is a valid arrangement: P T U Q S V R.
- (T U) (Q S V R) P: Here, Block 1 is next to Block 2, followed by P. This is also a valid arrangement: T U Q S V R P.
Let us check the statements in the options against our valid arrangements (e.g., P T U Q S V R):
- Q and R are not parked together: Looking at (Q S V R), S and V are between Q and R, so they are indeed not parked together. This statement is correct.
- V is the only car parked in between S and R: In (Q S V R), V is directly between S and R. This statement is correct.
- Car P is parked at extreme end: In the valid arrangements P T U Q S V R and T U Q S V R P, P is parked at the extreme left or extreme right end. This statement is correct.
- There are two cars parked in between Q and V: In the block (Q S V R), only S is parked between Q and V. This means there is only 1 car between them, not 2. Therefore, this statement is incorrect.
Thus, the only incorrect option is indeed "There are two cars parked in between Q and V."
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