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?

$100$

$125$

$600$

$2300$

$3600$

Answer: C

Concepts:combinations permutations arrangements with restrictions

Difficulty rating: 1700

Solution:

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.

2007 AMC 10B Problem 20

Answer: C

Difficulty rating: 1700

Solution:

Problem 20 in Other Years