Let us consider only the 1st 25 places. Convince yourself that the people beyond position 25 in the queue do not change our probability of winning a free drink.
For these 25 people, there are 365^25 different birthday assignments.
Now, the only 'birthday assignments' where we win a prize are where the following 2 conditions hold:
1)The 1st 24 people have distinct birthdays: (365,24)*24! ways
2)Our birthday is 1 of these 24: (24,1) ways
So, favorable birthday assignments are (365,24)*24!*24.
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