Adding one more person adds more than one comparison. Assuming birthdays are evenly distributed (they aren't), 2 people have a 1/365.25 chance of sharing a birthday. But adding a third person adds 2 more chances. Adding that 23rd person adds 22 more chances than adding the 22nd person. 1+2+3...+22=253 separate checks, of which only one needs to match.
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