How many ways are there to distribute five balls into seven boxes if each box must have at most one ball in it if?

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Katie M.

Calculus 3

1 year, 2 months ago

We don’t have your requested question, but here is a suggested video that might help.

How many ways are there to distribute five balls into three boxes if each box must have at least one ball in it if a) both the balls and boxes are labeled? b) the balls are labeled, but the boxes are unlabeled? c) the balls are unlabeled, but the boxes are labeled? d) both the balls and boxes are unlabeled?

Because at most, we can only have 1 ball in each box, we know that the partition we will be working with is $1+1+1+1+1$.

I know that if we take away the rule of at most 1 ball per box, the answer would be $8^5$ ways to distribute the balls to the boxes, but i'm not sure how to calculate this.

At first, my thought is to take $8^5$ and subtract every other scenario where a box has more than one ball, but I feel like that would take longer than just counting how many will have one each?

EDIT: The balls and boxes are both identical.