Problem:
How many distinguishable arrangements are there of brown tile, purple tile, green tiles, and yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let's first find how many possibilities there would be if they were all distinguishable, then divide out the ones we overcounted.
There are ways to order objects. However, since there are ways to switch the yellow tiles around without changing anything (since they're indistinguishable) and ways to order the green tiles, we have to divide out these possibilities:
There are choose ways to arrange the yellow tiles which is . Then from the remaining tiles there are ways to arrange the green tiles. And now from the remaining two tiles and two slots, we can see there are two ways to arrange the purple and brown tiles, giving us an answer of
The problems on this page are the property of the MAA's American Mathematics Competitions