Fishy distributions

Probability Level 1

In a fish tank, there are 4 distinct fish. You spread 10 identical food pellets into the tank.

How many distributions of food pellets among the fish are there?


The answer is 286.

Andy Hayes by Andy Hayes , 38, USA

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1 solution

Andy Hayes
May 12, 2016

Relevant wiki: Identical Objects into Distinct Bins

This problem can be modeled as n = 10 n=10 identical objects distributed among r = 4 r=4 distinct bins.

Use the formula for these kinds of distributions : ( n + r − 1 r − 1 ) = ( 13 3 ) = 286 \binom{n+r-1}{r-1}=\binom{13}{3}=286

Thus, there are 286 \boxed{286} possible distributions of the food pellets.

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How do you know when to use count stars and bars and when to use this formula?

Laurence Kuo - 3 years, 4 months ago

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Check out the wiki page that I linked on Identical Objects into Distinct Bins. We also have a wiki on stars and bars .

Andy Hayes - 3 years, 4 months ago

I'm guessing that you don't need require the answer anymore, given that it's been over 2 years since you asked this question, but I'll answer it anyways. The formula is a generalized result derived from the 'stars and bars' or, as I prefer to call it, the binary sequence method where no two 1's are kept together. I recommend using the method to solve questions while practicing, as it enables you to utilize the method in wide variety of questions, and not just for distribution. If it's an exam however, it would be quicker to use the formula.

Nakshat Pandey - 7 months, 2 weeks ago

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