2021 April MIMC 10 Problems/Problem 13
Given that Giant want to put green identical balls into different boxes such that each box contains at least two balls, and that no box can contain or more balls. Find the number of ways that Giant can accomplish this.
We can use stars and bars to calculate the total amount of ways without thinking about the condition "no box can contain or more balls". Giving each box ball in advance to meet "each box contains at least two balls" requirement since each box will get at least extra, There are or ways to accomplish this. We need to subtract the situations where a box contains or more balls. There are two situations: in each box or in each box. In the first situation, since two boxes have the same amount, it will have ways of accomplishing it, and the second situation will have ways of accomplishing it. Therefore, the total amount of ways that meets the requirement is .