Question

How many people must you have to guarantee that at least (9) of them will have birthdays in the same day of the week

Answer 1

Given,

At least 9 of the people will have birthdays on the same day of the week.

To Find,

A number of people should be there for sure to maintain the criteria.

Solution,

From the generalized form of the Extended pigeonhole principle, We can say that

the number of pigeonholes is smaller than the number of pigeons, then the extended pigeonhole principle applies. Like, If "n" be the number of pigeons and "m" be the number of pigeonholes, and if (m < < n)  the according to the extended pigeonhole principle, one pigeonhole must contain at least one pigeon $\frac{n-1}{m} +1$ .

 Here assume the pigeon holes to be the 7 days of a week and pigeons to be the people.

SO We need to find the number of pigeons = n =?

No. of pigeonholes = m = 7 (days).

So,  $\frac{n-1}{m} +1$=9.

⇒$\frac{n-1}{7}=9.$

⇒n-1=7×9.

⇒n=63+1=64.

No. of pigeons is 64.

Hence, 64 friends should be there to guarantee that at least 9 of them must have a birthday on the same day of a week.