Question

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.)

$210$

$420$

$630$

$840$

$1050$

Answer: B

Concepts:multiset permutations

Difficulty rating: 900

Small Hint:

First arrange the seven tiles as if they were all different

Big Hint:

Divide by the orders of the identical green and yellow tiles

Solution:

There are $1+1+2+3=7$ total tiles. If all seven tiles were distinct, there would be $7!$ arrangements. The two green tiles are indistinguishable, and the three yellow tiles are indistinguishable, so we divide by $2!$ and $3!.$

Thus the number of arrangements is $\frac{7!}{2!3!}=420.$

Thus, B is the correct answer.

2020 AMC 10B Problem 5

Answer: B

Difficulty rating: 900

Small Hint:

Big Hint:

Solution:

Problem 5 in Other Years