Question

How many people need to be in a group to claim that at least 3 people will have the same birthday? (account leap year)

Answer 1

Before start of this question ! we have to know about a simple concept of pigeon hole principle .

* then what is pigeon hole principle ?

this principle states that if there are (N) numbers of holes and (N+1) or more then (N+1) numbers of Pigeons , and all the pigeons are placed in these holes then principle states states that atleast one of the holes must contain more than one pigeon in it .

Now We Can Take a example for better understanding !

lets say holes are 10 and pigeons are 11 then 10 holes will fill up by 10 pigeons and the last pigeon that is 11th pigeon will definitely placed with any of the previously occupied hole !

So Now Focus On Our Problem :

As Here it is asking , number of persons in a group so that 3 of them can claim birthday on same date , and also asking for considering leap year that is( 366 days )

So Lets Say If there are 366 persons then lets say

1st person will have birthday on 1 January

2nd person will have birthday on 2nd January

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366th person will have birthday on 31 december or last day of year that is 366th day !

( note :- this is crtical case of having distinct birthday for all people , on different date , may be two of them or more have same birthday but in that case definitely there will be some dates on which no one will have birthday ! so we are taking this critical case of having distinct birthday for all 366 people! )

Now All days have been filled of anybody's date of birth .

if we add one more person in the group than definitely we can say that atleast two of them can claim birthday on same date !

Now if we add 366 more persons in the group means total persons = 366+366= 732 persons

then we can observe one case : that each day of year is basically birhday of two persons

Note :- here someone can say that if we just add 2 more members with 366 members instead of adding 366 members then also there can be 3 persons who can claim birthday on same day ? ......but try to understand that in this case we can not say definitely that all the three persons was born on same date ! may be they have birthday on different dates !

- so just after adding 366 more members so total become 732 , and if we add just a one more member then definitely we can say that there will be three persons who can claim of having same birthday as because here all the date are filled twice . so if we just add one more then definitely one of the date should be same for three members Birthday ,

So There should be 732+1= 733 members

so if there are 733 members or more than 733 members in a group then among those members atleast three members will have same birthday !

so answer is 733 peoples !

Thanku :)

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