Pigeon hole exercise

Question

I have an exercise that it is pigeon hole principle.I have to ask you if i solve it right. The question is : There is a university, the students get in the univeristy ) from the 2003 year until 2018.How many at least students ,must come in a party that university does,if there must be for sure at least 10 students(the same year) ? What i did is i count years from 2003 until 2018 and it is 15 years so i did the pigeon hole principle i got the 15 years : 10 at least students ,that i did [15/10]=[1.5] =2

Answer 1

Imagine you are at the door of the party and choosing who to let in. You are trying to avoid letting in $10$ people from the same year at university. You can then let in nine people from 2003, but not a tenth. Similarly, you can let in nine people from 2004, but not a tenth. The same goes for all the other years too, so you can let in $9$ people from each of the years 2003 to 2018. That is $9\times16=144$ people, because there are $16$ years (2003 and 2018 are both included).

If you now let a $145$th person in, they must be from one of the years 2003 to 2018, and therefore must complete a group of ten from the same year. So it seems $145$ is the smallest party for which you are sure to have $10$ people from the same year.

You can prove it with the (generalized) pigeonhole principle as follows:
If you split $145$ university people into their $16$ year groups, the average group size is $145/16 = 9+\frac{1}{16}$. Therefore there must be at least one group with more than $9$ people (i.e. $10$ or more).