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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!

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    $\begingroup$ I assume you meant to say each box can contain one ball. Hint: Select which boxes will receive a ball. $\endgroup$ May 12, 2021 at 1:58
  • $\begingroup$ Yes, I meant that a box can be either empty or occupied by 1 ball. $\endgroup$
    – Light
    May 12, 2021 at 2:07
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    $\begingroup$ The answer is $\binom{y}{x}$ (Im assuming $y$ is at least equal to $x$ as otherwise there is no way to do it.) $\endgroup$
    – Asinomás
    May 12, 2021 at 2:12
  • $\begingroup$ Thank you for your answer! This results makes sense in the context of my problem. $\endgroup$
    – Light
    May 12, 2021 at 3:02
  • $\begingroup$ By convention, $\binom{y}{x} = 0$ if $y < x$. Thus, the answer is $\binom{y}{x}$ regardless of whether it is physically possible to distribute the balls to the boxes with the stated restriction that no more than one ball can be placed in each box. $\endgroup$ May 12, 2021 at 11:33

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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.

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