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# counting

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Please help with this counting problem: 16 pebbles are arranged in a 4 by 4 array.  Find the number of way of choosing four pebbles, so that no two of the chosen pebbles lie in the same row or column.

May 10, 2020

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+24992
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Find the number of way of choosing four pebbles,
so that no two of the chosen pebbles lie in the same row or column.

My attempt:

16 choices for a first pebble,
9 choices for a second pebble, and
4 choices for a third pebble, and
1 choice for a fourth pebble gives a total of $$16\times9\times4\times1 = 576$$, if the pebbles are numbered,
when not, gives a total of $$\dfrac{16\times9\times4\times1}{4!} = 24$$

By rotation there are 2 different  possibilities:

 x x x x
 x x x x
 x x x x
 x x x x

By rotation there are 4 different  possibilities:

 x x x x
 x x x x
 x x x x
 x x x x

May 11, 2020