Hi so I'm having trouble understanding the difference between Bose-Einstein theorem, and multinomial coefficient and when to use them separately?
So the problem I had to work on was "a group of 30 dice are thrown how many different ways are there for 5 of each of the values to appear on a 6-sided die (1,2,3,4,5,6) thrown 30 times?"
so the multinomial coefficient , is is counting the number of sides (1-6) which appear in 30 tosses. $\frac{30!}{5!^{6}}$.
This is equivalent of saying 30! total ways of ordering the dice, however we do not care about the order that the 5 sides appear and must correct for the 5! ways a given face appears since we do not care about the ordering of a particular face/side.
however the Bose-Einstein formula, is counting the total number of $k$ indistinguishable particles which are within $n$ distinguishable 'bins'. So for example, in this case if each side of the die was a distinguishable bin (1,2,3,4,5,6), and we are interested in the 'tallying up' how many times a bin was selected in 30 tosses, won't this yield $35\choose{30}$ total ways of distributing the 30 tosses across bins/faces ? yet these are very different!
please help me understand the differences between these approaches.
The dice play the role of balls and the sides play the role of cells. There are $r=30$ dice throws which could land any of the values $1,2,3,4,5,6$ i.e. $n=6$ cells. We choose to treat the dice as distinguishable, simply because the sequence $1111122222333334444455555$ is distinct from $5555544444333332222211111$.
There are $30!$ permutations of $30$ distinct dice. As you pointed out, if we take any sequence for instance $111112222233333444445555566666$, corresponding to each sequence there are $(5!)^6$ arrangements that leave the outer appearance of the sequence unchanged. Hence, the division by $(5!)^6$. So, $\frac{30!}{5!5!5!5!5!5!}$ are the number of distinguishable arrangements of five 1's, five 2's, five 3's, five 4's, five 5's and five 6's.