parveen110 wrote:In how many ways is it possible to choose two black squares on a 8 × 8 chessboard so that the squares do not lie in the same row or same column?

a. 400
b. 200
c. 800
d. 100
e. 475

I solved this question using the same technique as Mitch's.
However, here's another approach that works:

First IGNORE the rule that says the squares must not lie in the same row or same column.
So, we need only count the number of ways that we can select 2 black squares from 32 black squares.
Since the order of the selected squares doesn't matter, we can use combinations.
We can select 2 squares from 32 squares in 32C2 ways
32C2 = (32)(31)/2 = (16)(31)

IMPORTANT: we need not evaluate (16)(31). You'll see why shortly.

Aside: If anyone is interested, we have a free video on calculating combinations (like 32C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Since we have IGNORED the rule about squares not lying in the same row or same column, our total of (16)(31) possibilities includes some scenarios that BREAK the rule and some scenarios that FOLLOW the rule. We're going to determine the PROPORTION of scenarios that FOLLOW the rule.

PROPORTION of scenarios that FOLLOW the rule
I'll borrow Mitch's diagram for this.

Once we select 1 black square (like the square highlighted in blue above), there are 31 black squares remaining. Of those 31 black squares, 6 BREAK the rule and 25 FOLLOW the rule (about squares not lying in the same row or same column).
In other words, for each of the (16)(31) possibilities that we calculated above, 25/31 of them FOLLOW the rule.

So, the # of possibilities that FOLLOW the rule = (16)(31)(25/31) = (16)(25) = [spoiler]400 = A[/spoiler]

Cheers,
Brent

HI parveen110,

Under normal circumstances, I would approach this question in much the same way that Mitch approached it.

However, this question has an interesting "design shortcut" to it that would help you to answer it quickly if you were low on time.

Since the chessboard has half white squares and half black squares, we know that there are 32 of each. When selecting 2 black squares (regardless of the limitations), the first "part" of the calculation is 32....

32 x (something)

Also, since we're asked for the possible ways to choose 2 black squares, that implies that we're dealing with "combinations" of squares, so we'll have to divide by 2 to remove the duplicates.

The ONLY answer that's evenly divisible by 32 is 800; THAT number, when divided by 2 is 400. If I was pressed for time, then I'd have to choose A.

GMAT assassins aren't born, they're made,
Rich