Count The Number of Arrangements

There are 10 parking spaces numbered from 101 to 110. At least one car is parked in these slots. If cars can be parked only at the consecutively numbered parking slots, how many such arrangements can be made. 

Consider that only one car can be parked in one parking slot and all cars are identical.

Here is the possible count! 

Possible Number of Arrangements

What was the puzzle?

Suppose there is only 1 car that is to be parked in 1 of the 10 slots. 

Number of possible arrangement = 10C1 = 10!/1!9 = 10.

That is 1 car can be parked in 10 slots in 10 number of ways.

Now, let's suppose that there are 2 cars that to be parked in 2 of the 10 parking slots. But the condition is that they need to be parked in consecutive slots. 

Among 10 slots for there are 9 possible consecutive slots for 2 cars. That is, 2 cars can be parked in consecutive slots in 9C1 = 9 number of ways. It's like placing 1 group of cars (having 2 cars) in 9 possible slots.

Similarly, in 10 parking slots for parking 3 cars there are 8 possible consecutive slots. Hence, there are 8 such arrangements are possible.

And so on for the rest number of cars.

Hence, there are total 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55 such arrangements are possible.  

Find The Parking Floor

Six people park their car in an underground parking of a store. 

The store has six floors in all. Each one of them goes to different floor.

Simon stays in the lift for the longest.

Sia gets out before Peter but after Tracy.

The first one to get out is Harold.

Debra leaves after Tracy who gets out at the third floor.

Can you find out who leaves the lift at which floor? 

 
Was it really so difficult? 

Finding The Parking Floor

What was the given data? 

The sentence 'The first one to get out is Harold.' suggests that the Harold leaves the lift at the 1st floor.

Similarly, the sentence - 'Simon stays in the lift for the longest.' suggests that Simon leaves he lift at the 6th floor.

Next, sentence - 'Debra leaves after Tracy who gets out at the third floor' suggests Debra leaves the lift at the 3rd floor.

Clearly, the sentence - 'Sia gets out before Peter but after Tracy.' suggests that Sia leaves the lift at the 4th floor.

Also, it is clear that Debra leaves the lift at the 2nd floor and Peter at the 5th floor.

Summary -

Harold leaves at the first floor.

Debra leaves at the second floor.

Tracy leaves at the third floor.

Sia leaves at the fourth floor.

Peter leaves at the fifth floor.

Simon leaves at the sixth floor.