Question

If 4 squares are chosen at random on a chessboard (there are 64 squares arranged in 8 rows and 8 columns in a chessboard), then the probability that all the four squares are in the same main diagonal is:

Answer 1

If I am not missing anything then,\[\frac {\binom 8 4}{\binom{64}{4}}\]

Answer 2

I would like to know How

Answer 3

There are 8 squares in the diagonals so choosing 4 from 8 and all possible one is choosing 4 from 64.

Answer 4

Well, no. of ways of choosing four squares outta 64---> 64C4 ---> sample space
Squares constituting the each diagonal -->8
and two diagonals --> so two cases.
choosing 4 squares out of the 8 squares on each diagonals = 8C4 ways

Now since there are two diagonals, there can be 2*(8C4) ways of getting the required condition.



So, P = [2*(8C4)]/[64C4]

Answer 5

uh-oh @FoolForMath seems you forgot, there are 'two' main diagonals.

Answer 6

But it says "the same main diagonal"

Answer 7

The other one is known as off-diagonal or does linear algebra is befuddling me?

Answer 8

Well, that fine, once for the diagonal1, and second for the diagonal2

so this way 'and' that way.
so we add 'em up.

thats what I think, though I don't stand a chance before you.

Answer 9

*that's

Is there an 'off' diagonal term for the chess boards? I am not sure. I guess main diagonal just refers to the two 'longest' diagonals.

Answer 10

Chess ref:

http://chessprogramming.wikispaces.com/Diagonals
http://chessprogramming.wikispaces.com/Anti-Diagonals

Linear algebra ref:

http://en.wikipedia.org/wiki/Main_diagonal

Answer 11

"However, we define the main diagonal on the chess board from a1/h8 and the main anti-diagonal from h1\a8"

Answer 12

I'm lost !

Answer 13

What does it mean by "all the four squares are in the same main diagonal " ?

Answer 14

I know, ISAT -2011 isn't? (I just googled)

Answer 15

and they gave both in the options.

Answer 16

Yup!

Answer 17

So which is the right answer ?

Answer 18

Okay Thanks!!

Answer 19

Yeah ISAT-2011 (that's why this problem seemed familiar)
Both answers were mentioned correct, because every high-school level student may not be the familiar with the terminology in Chess, and may assume 'main diagonals' to be any of the two diagonals.(basically what happened to me)


@Diyadiya , "all the four squares are in the same main diagonal " this means that when you choose 4 squares, all of 'em must lie on a diagonal. (and the same diagonal means the you can choose 'em from only any one of em, since there are two diagonals -thinking my way)

Like this.