Lost Cars
There is a parking lot with a wall on each end, and 10 side-by-side spaces in between. Ten cars park in these spots: 5 identical blue cars, and 5 different red cars. A blue car owner knows where he is parked if 1) he is next to a red car, or 2) he is on the far left or right (because he knows he is next to the wall). How many different arrangements of red and blue cars can there be in which every blue car knows where he is parked?
(Note: Although the red cars are unique, for the sake of combinations, count them as the same. For example, if there were two different red cars, there would only be 1 combination, RR)
The answer is 126.
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1 solution
Since there are 5 red cars, we can say that there are 6 spaces that blue cars can go: on the far left, the four spaces in between, and the far right. And none of these spaces can have more than 2 blue cars, because if there are 3 blue cars, the one in the middle does not know where he is parked. So we have to determine how many will contain 1 car, and how many will contain 2.
Cases:
1: If there are five spaces with 1, there are 6 ways to do this, one for each space that could be empty.
2: If there are 3 spaces with 1 and 1 space with 2, there are 6 ways to place the two TIMES 5c3, or 10, ways to place the remaining three, so 60 cases.
3: If there is 1 space with 1 and 2 spaces with 2, there are 6 ways to place the one TIMES 5c2, or 10, ways to place the remaining two, so 60 cases.
This gives us 126 ways.