600 identical balls are distributed randomly in 6 numbered boxes (all balls are to be distributed and boxes can be empty). What is the probability that exactly 300 balls will end up in the first 3 boxes?

600 identical balls are distributed randomly in 6 numbered boxes (all balls are to be distributed and boxes can be empty). What is the probability that exactly 300 balls will end up in the first 3 boxes?

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Answers have to be complete and motivated.
The answers can contain factorials, binomial numbers, Sterling numbers and +-*/
1)
600 identical balls are distributed randomly in 6 numbered boxes (all balls are to be
distributed and boxes can be empty). What is the probability that exactly 300 balls will
end up in the first 3 boxes?

Expert Solution

Step 1

When there are $n$ identical objects to be distributed among r distinct bins, we will have $(r - 1)$ bars that divide them into $r$ distinct groups. The total objects that will be distributed are $(n + r - 1)$ , and $(r - 1)$ of these distributions are made for the bars alone. Thus, there are $(\overset{n + r - 1}{r - 1})$ possible placements of the bars.

Equivalently, there are $(\overset{n + r - 1}{r - 1})$ possible distributions of n identical objects among r distinct bins.

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