Pigeonhole principle: Asking the minimum number of students 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.

.
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).
Think of it this way: if you have $4950$ students total, then that could mean you have exactly $99$ from each state. To guarantee that one state has $100$ students, add one more.