Consider a flexible form of bingo, where each square contains a condition and you mark off whether or not the condition applies to you. The number of bingos you obtain ostensibly measures the extent to which you satisfy the theme of the bingo card. I, a soulless automaton, have inferred that these are commonly shared as images on social media as ways to bond over shared preferences.
Suppose, in my purely hypothetical robotic misanthropy, I wished to construct a standard bingo card (5x5 with a free space in the center) on which no bingos (vertical, horizontal, or diagonal) were possible. To do this, on each possible bingo line, place a pair of squares which cannot be simultaneously satisfied, such as "tall" and "short". Twelve possible bingos mandates twelve such pairs of squares, and luckily enough there are 24 usable squares in a standard bingo card.
Here is an example of one such malicious bingo card, using the letters A through L to denote the pairs:
AABCD EEBCD FG_HH FIJKJ IGLLK
How many malicious bingo cards are there? Is there a human-understandable way to count them, or should this enumeration be left to a computer like myself?
On a secondary note, are there well-known combinatorial objects they are in a natural bijection with? Searching the vast indices in my hard disks, I can only come up with "maximum matchings in a particular 24-vertex graph such that each maximal clique contains exactly one matched edge"—I do not expect to find something this convoluted in any of your human mathematical literature.