2007 AMC 10B Problem 20

Question

A set of

25

square blocks is arranged into a

5\times 5

square. How many different combinations of

3

blocks can be selected from that set so that no two are in the same row or column?

100

125

600

2300

3600

Choose which

3

rows and which

3

columns are used

After picking

3

rows and

3

columns, the blocks correspond to a matching between them

Choose

3

of the

5

rows in

\binom{5}{3}=10

ways and

3

of the

5

columns in

\binom{5}{3}=10

ways.

The three chosen blocks must occupy distinct rows and columns, so they form a matching between the three rows and three columns, which can be done in

3!=6

ways.

The total is

10\cdot 10\cdot 6=600.

Thus, the correct answer is C.

Problem 20 in other years