Permutation is the act of arranging the members of a set into a sequence or order, or, if the set is already ordered, rearranging (reordering) its elements—a process called permuting.
How many distinguishable arrangements are there of 1 brown tile,1 purple tile ,2 green tiles and 3 yellow tiles in a row from left to right ? (Tiles of the same color are indistinguishable.)
A) 210 B) 420 C) 630 D) 840 E) 1050
American Mathematics Competition 10 (AMC 10B), 2020, Problem Number - 5
Permutation
5 out of 10
Mathematical Circle
First hint
If you really need a hint you can go through the concept of probability at first : In its simplest form, probability can be expressed mathematically as: the number of occurrences of a targeted event divided by the number of occurrences plus the number of failures of occurrences (this adds up to the total of possible outcomes):
\(p(a) = \frac {p(a)} { [P(a) + p(b)] } \)
Second Hint
Let's try to find how many possibilities there would be if they were all distinguishable, then divide out the ones we over counted . There are 7! ways to order 7 objects. However, since there's 3!= 6 ways to switch the yellow tiles around without changing anything (since they're indistinguishable) and 2! = 2 ways for green tiles.
Final Step
I am sure that you are almost there for the final calculation but let me help those who are still not there
\(\frac {7!}{6 \cdot 2} = 420 \) .
So the correct answer is B) 420
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