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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
 #1
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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.

 

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   

 

laugh

 May 11, 2020

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