Pigeonhole principle: Asking the minimum number of students

Question

The question

What's the minimum number of students, each of whom comes from one of the 50 states must be enrolled in a university to guarantee that there are at least 100 who come from the same state?

The solution

50*99 + 1 = 4951.
4951/50 = 100

My question

I don't understand where 99+1 comes from? 100? and why do we have to formulate 100 to 99+1? Thank you.

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Answer 1

If you take $50\times 99$ students ($99$ by state), you are sure to have at least $99$ students coming from each state. If you add $1$, whatever the state he comes from, you are sure there are $100$ students coming from the same state (and there is only one state from which $100$ students come).

Answer 2

If you enroll $50 \times 99$ students, your goal might still not be reached, because of the extreme case that you have an equal distribution of states up to this point, thus $99$ students from each of the $50$ states.

If you enroll one more student he or she must complete one state fraction to at least $100$ students and you can be sure of your goal.

Answer 3

By extended pigeon hole... n = pigeon m = hole

First you have to calculate. (n-1)/m then add 1 to result..

Now if you compare this question we have to find students (pigeon) and State are hole and given that at least 100 come from same state so our ans is 100... If you see previous after divide (n-1)/m then add 1 to result..

now if you reverse the process so you need to first substract 1 from results (it is 100-1 = 99 ) then multiply by 50 (m = 50) ( 99×50) then again add 1 which is(n-1 so 1 go right side so it's Plus) ... So answer is n = 4951.. Hope you clear