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Plz help with counting

 

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?

 Feb 6, 2023
 #1
avatar+118617 
+1

ABCDEFG  that is 7 friends

Cut into 9 peices that is 8 bars.     

7+8=15

15C8=6435   ways to distribute.

 

If D gets 2 peices than you can take him and the 2 choc he is getting out  

then you are left with

 

ABCEFG   6 friends    divided into 7 parts that is 6 bars

6+6=12

12C6 = 924 possible ways

 

so that is    924/6435 =         28 / 195

 

I used the Stars and bars method to solve it. 

I could have make a logic error. You need to decide for yourelf whether to accept this answer.

 Feb 6, 2023
 #2
avatar+2667 
+1

There are 9 stars (chocolate) and 7 - 1 = 6 bars. 

 

This makes for \({15 \choose 6} = 5005\) cases. 

 

If Dhruve gets exactly 2, there are now 7 stars, and 6 - 1 = 5 bars, which makes for \({12 \choose 5} = 792\) cases. 

 

So, the probability is \({792 \over 5005} = \color{brown}\boxed{72 \over 455}\)

 Feb 6, 2023

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