I saw What is the expected value of the largest of the three dice rolls? and decided to go with a different approach than the ones that were mentioned. I was wondering if someone can point out why my approach is incorrect.
Using the stars and bars method, the number of ways s.t. the max is 6 is
$$ {{2 balls + 6 bins - 1} \choose 2 balls} = 21 $$
The reason I use two balls is because I already put one in bin 6, to ensure 6 is the max. The other two balls can be put into any of the 6 bins.
Using similar logic,
# of ways s.t. max is 5: ${{2 balls + 5 bins - 1} \choose 2 balls} = 15$
# of ways s.t. max is 4: ${{2 balls + 4 bins - 1} \choose 2 balls} = 10$
# of ways s.t. max is 3: ${{2 balls + 3 bins - 1} \choose 2 balls} = 6$
# of ways s.t. max is 2: ${{2 balls + 2 bins - 1} \choose 2 balls} = 3$
# of ways s.t. max is 1: ${{2 balls + 1 bins - 1} \choose 2 balls} = 1$
Total # of ways: ${{3 balls + 6 bins - 1} \choose 3 balls} = 56$
We thus get an expected value of $\frac{126 + 75 + 40 + 18 + 6 + 1}{56} = \frac{19}{4}$, which is incorrect. What part of my logic is wrong?