Show that among a group of 621 people there are at least 21

Show that among a group of 621 people, there are at least 21 who are born on the same day of the month (e.g., the 21^st or the 12^th, etc .). Is the same fact true if there are only 620 people?

Solution

Assuming there are 31 days in the month

So there are at least 20 born on same day if we can say assign one day to each and hence that would account for :20*31=620 people

So if the remaining one person is assigned one day of the month he will have common birthday with 20 other people and hence 21 people would hv same birthday

Hence proved

No it is not true for 620 as we proved above as we can assign a day of birth to 20 people for each day of the month and hence 20 people would shared the birthday for each day of the month