Question

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

Short answer

\(4951\) students.

Step-by-step solution

Given data

Each student comes from one of the \(99 \times 50 = 4950\) states.

Concept used of number of ways to do a task

If a task is complete in\(n1\)ways or\(n2\) ways, then the number of ways to do the task is\(n1 + n2\).

Find the number of students

Each student comes from one of the \(50\) states.

We take the least case where we consider to have \(99\) students from each state.

So there are \(99 \times 50 = 4950\) students in the university.

The next final student is the one from any of these \(50\) states so that he is the \({100^{{\rm{th }}}}\) student from that specific state.

Thus, if \(4950 + 1 = 4951\) students are considered, such that there is at least one student from each state, then there is at least one state from which \(100\) students might have come.