Question 31
A calendar is a rectangular sheet of grid paper. We call a calendar *legal* if and only if it satisfies the following three conditions:
(1) Every cell of the calendar is colored white or red, and there are exactly $10$ red cells.
(2) If each row of the calendar has $N$ cells, then when we write $1, 2, \dots$ into the cells starting from the top-left cell, from left to right and then from top to bottom, we cannot find $N$ consecutive numbers whose cells are all white.
(3) If each column of the calendar has $M$ cells, then when we write $1, 2, \dots$ into the cells starting from the bottom-left cell, from bottom to top and then from left to right, we cannot find $M$ consecutive numbers whose cells are all white (in other words, if we rotate the whole calendar $90^\circ$ clockwise, it still satisfies condition (2)).
How many legal calendars are there?
(1) Every cell of the calendar is colored white or red, and there are exactly $10$ red cells.
(2) If each row of the calendar has $N$ cells, then when we write $1, 2, \dots$ into the cells starting from the top-left cell, from left to right and then from top to bottom, we cannot find $N$ consecutive numbers whose cells are all white.
(3) If each column of the calendar has $M$ cells, then when we write $1, 2, \dots$ into the cells starting from the bottom-left cell, from bottom to top and then from left to right, we cannot find $M$ consecutive numbers whose cells are all white (in other words, if we rotate the whole calendar $90^\circ$ clockwise, it still satisfies condition (2)).
How many legal calendars are there?