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Miyu is giving out 9 identical chocolates to her 7 friends, including Dhruv.  All possible distributions are equally likely.  What is the probability that Dhruv gets exactly 2 chocolates?

 Jan 1, 2023
 #1
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First we use Stars and bars to find all possible cases:

\({x}_{1}+{x}_{2}+{x}_{3}...+{x}_{7}=9\)

Stars and bars tells us that there are (9+7-1) choose (7-1) = 5005 nonnegative integer solutions to this.

Now we assume Dhruv (assume that he is x7) gets 2 chocolates. We can find the number of cases which Dhruv gets 2 chocolates. 

\({x}_{1}+{x}_{2}...+{x}_{6} = 7\)

Now we use stars and bars again to find that there are (7+6-1) choose (6-1) = 792 nonnegative integer solutions.

So the probability that Dhruv gets 2 chocolates is (792/5005) = (72/455)

(There might be a calculation error if there is tell me)

 Jan 1, 2023
 #2
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That answer is wrong.

Guest Jan 1, 2023

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