What is the sample space for randomly placing 8 pieces on an 8x8 checkerboard?

[playboy]

The question asks:

Suppose that we randomly place 8 playing spieces on an 8 x 8 checkerboard.
What is the probability that each row of the checkerboard will have only one piece?

I've been trying forever to get this question going. My problem is that I cannot figure out the magnitude of the sample space. Can somebody please help me out with this?

I tried to find the sample space for a 2x2 and 3x3 board.

For a 2x2 board i got |S| = 6
For a 3x3 board i got |S| = 24
For a 4x4 board...it got really messy...

Can somebody give me a boost please!?

 

[FunkyDwarf]

well if there's 8x8 then the sample space is 8^2

 

[playboy]

unless i did it wrong, for a 2x2 board 2^2 = 4 and not 6
and for a 3x3 board, 3^2 = 9 and not 24

 

[FunkyDwarf]

well if you think about it...

1 x x x
2 x x x
3 x x x
1 2 3

 

[playboy]

This is how i interpret it:

. . .
x x x
x x x

x x x
. . .
x x x

x x x
x x x
. . .

. x x
x . .
x x x

. x x
x x x
x . .

x . .
. x x
x x x

x x x
. x x
. . x

etc. etc..

Which is more than 3^2 = 9

What do you say?

 

[FunkyDwarf]

i say I am not sure :P the sample space is still just the arena of outcomes, but your listing all the different outcomes from one sample space. its like the dice example, you have a 6x6 sample space but you can choose any number of combinations, but the sample space is still 6x6

 

[playboy]

So can somebody please confirm what the sample space is?