1. The problem statement, all variables and given/known data In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let be the number of shadings with this property. Find the remainder when is divided by 1000. There is a picture at this link if you do not understand the question: http://www.artofproblemsolving.com/Wiki/index.php/2007_AIME_I_Problems#Problem_8 2. Relevant equations 3. The attempt at a solution So, somehow we need to divide this into case or find some equivalent problem or something. Symmetry might help us since reflection about the middle horizontal line and reflection about the middle vertical line will preserve the property. Also, inversion of color will preserve the property. So the answer (before you take the remainder) must be a multiple of two. Hmmm...I am out of ideas.