After school hours, all students will take their bikes and pay 1 dollar. Supposing that there are m students with 1-dollar bills, k students with 2-dollar bills (every student has either 1-dollar or 2-dollar, there is no one with both) and the keeper has no cash at the start. How many ways can all students take their bikes out without the keeper having to go out to exchange money?
My approach is that every time, the number of students with 1-dollar bills that have already taken their bikes is higher than those with 2-dollar bills.
Now, for every 1-dollar bill student, I will mark them with 0 and the other 1 to make a binary string. The question now is how many binary strings satisfy the requirements of the problem.
Can anyone help me to progress on this problem?
