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?

Guest Feb 6, 2023

#1**+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.

Melody Feb 6, 2023

#2**+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 \({792 \over 5005} = \color{brown}\boxed{72 \over 455}\)

BuilderBoi Feb 6, 2023