Probabilty that 2 people in the group dont share a birthday
=(365/365)*(364/365)
Probabilty that 3 people in the group dont share a birthday
=(365/365)*(364/365)*(363/365)
Probabilty that n people dont share a birthday
(365!)/((365 - n)!*(365^n)) = P(no shared birthday)
By substitution we can see that for n = 23 the probability just drops below 0.5 and since the required probability is
1 - P(no shared birthday) = P(shared birthday)