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# Counting

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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
+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
+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