In order to explain some physics related experimental results, I would have to find a general formula for the following problem:
"How many different ways are there to put $x$ indistinguishable balls into $y$ distinguishable boxes, given that every ball has to be put inside a box and each box is either left empty or receives exactly one ball."
Can anybody provide some help with this. Thanks!
Best Answer
Each box here can contain either 0 or 1 ball, and there are $y$ boxes. This fact can be represented using binomial coefficients. We can rephrase the question as: $(1+t)(1+t)....(1+t)$ where $(1+t)$ has been multiplied $y$ times. What is the coefficient of $t^x$? Clearly the answer is $\binom {y}{x}$. Here, $(1+t)$ has been used for each box, because the power of $t$ represents the number of balls in that box. Thus, multiplying them together gives all possible combinations.