2020 AMC 10B Problem 5

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

First arrange the seven tiles as if they were all different

Divide by the orders of the identical green and yellow tiles

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.

Problem 5 in other years