Every route across a five by five grid is ten steps long with exactly five going east, so counting routes is choosing which five of the ten slots are east. The reflex 1024 is the exact number of free ten-step walks, and only 252 of them arrive. Forbid the route to rise above the diagonal and the count collapses to the Catalan number 42.
Four over fifty-two squared is exactly right for the question where the first card goes back, which is what makes it hard to catch. Removing a king shrinks the numerator proportionally more than the denominator, and the counting route through 1326 two-card hands confirms one in 221 without mentioning order at all.
The reflex answer is around 180, half the calendar. The real threshold is 23, because a match needs a pair and 23 people carry 253 of them. The same reasoning puts the answer to "does anyone share MY birthday" at 253 people, eleven times the crowd, and the square-root threshold behind both is why a 64-bit random id collides after five billion draws rather than eighteen quintillion.