Find the number of ways to park cars next to each other in a parking area

Question

Question is;

Find the number of ways to park $4$ cars next to each other in a parking area with $10$ distinct places to park if:

(A) The parking area is in the shape of a circle.

(B) The parking area is in the form of a rows.

Given answers: (A)$240$ (B)$168$

How do I solve it and what is the concept behind the question?

Answer 1

If the parking area is a circle, there are $4!$ ways to have the cars permuted. As the cars must all be parked next to each other, they form a "block" of $4$ cars. Pick an arbitrary car. There are $10$ possible positions it can take in the circle. Thus, the amount of ways is $4! \cdot 10 = 240$.

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If the parking area is a row of $10$, again there is $4!$ ways to permute the order of the cars. The cars form a contiguous "block" of $4$. Pick the leftmost car. It may be in positions $1,2,3,4,5,6,7$(it can't be $8,9,10$ because there are $3$ cars to the right of it). These give $7$ total placements of the block of four. Thus, the amount of ways is $4! \cdot 7 = 168$.

Answer 2

Here are some hints.

• How many places can you place four cars next to each other in each configuration? (A picture may help a lot.)
• For each one of those places, how many ways can you arrange the four cars?

Can you take it from here?