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P**ermutation** 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 elementsâ€”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*

Source

Competency

Difficulty

Suggested Book

*American Mathematics Competition 10 (AMC 10B), 2020, Problem Number - 5*

** P**ermutation

*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*

- https://www.cheenta.com/average-problem-from-amc-10a-2020-problem-no-6/
- https://www.youtube.com/watch?v=LLqmnHUtLoA

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