Math CentralQuandaries & Queries


Question from bobbym:


Four dots are randomly placed on an 8x8 grid, compute the probability
that no row or column contains more than one dot. My solution consists of saying that this
is modeled by rook polynomials. We just need the number of ways to place 4
non attacking rooks on an 8 x 8 chessboard. Using the well known rook
polynomial for an 8 x 8 board,

40320 x^8 + 322560 x^7 + 564480 x^6 + 376320 x^5 + 117600 x^4 + 18816 x^3 + 1568 x^2 + 64 x + 1
and checking the coefficient of x^4; So I have:

117600 / ncr(64,4) = 350 / 1891 as the answer. I think I am correct while other posters think I am wrong. What do you think?

I get the same answer. Instead of rook polynomials, I compute
(8*8*7*7*6*6*5*5)/4*3*2*1 = (117600) for the number of ways to place 4 non-attacking rooks on the chessboard.

There is a way to interpret the question differently and get another
answer: 117600/(644) if we don't assume that the dots will be placed in different squares.


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