Question

Three $\text{A's},$ three $\text{B's},$ and three $\text{C's}$ are placed in the nine spaces so that each row and column contain one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible?

$\ 2$

$\ 3$

$\ 4$

$\ 5$

$\ 6$

Answer: C

Concepts:arrangements with restrictions casework

Difficulty rating: 1240

Solution:

The other two A's must occupy one square in each of the remaining two rows and columns. There are two possible diagonal patterns for the three A's.

For either A-pattern, the two remaining positions in the top row can be filled as B,C or C,B, and then the rest of the grid is forced.

So there are $2\cdot2=4$ arrangements.

Thus, C is the correct answer.

2008 AMC 8 Problem 14

Answer: C

Difficulty rating: 1240

Solution:

Problem 14 in Other Years