Number of ways in the bomb game There are 8 people(namely A, B, ..., and H) and 8 bombs with their names on it(namely bomb A, bomb B, ..., and bomb H). There is a counter for each bomb.
8 people sit in the circle, holding 1 bomb each. When the bell rings, they pass the bomb to the person left. If a person receives his/her bomb, one of two happens:


*

*The counter of the bomb decreases by 1 if the counter is positive integer.

*The bomb explodes, removing the person if the counter is 0.


(If a person receives a bomb but not his/hers, nothing happens)
If a person is removed, then they 'connect' the circle. That is, suppose B was left to A and C was left to B. If B is removed, then the person left to A becomes C.
After the bell rings 30 times, all person must be removed. How many ways are there to set the counter and give the (name-tagged) bombs?
I tried to solve this, but couldn't know how. Any ideas?
 A: Assume a recording clerk logs the events, where each log entry is of the form "The bell rings and the players ... explode", where the "..." is a finite subset of the original players and the subsets are pairwise disjoint but together contain all original $n$ players. And of course we have $t$ lines in or log. (In the concrete OP example, we have $n=8$ and $t=30$).
From any  log of this form, we can reconstruct the original setting in a unique way as follows:
Arrange all players in their starting position, holding a dummy bomb (so that no actual explosions occur during our simulations) in their hand. The dummy bombs do not have player names on them but different colours. Replay the log, that is, with each "The bell rings" all bombs are handed one step to the left, and "exploding" players leave the circle with their dummy bombs. In the end, everybody writes their name on the dummy bomb they hold.
Now arrange all players in their starting position again, holding the dummy bomb of the same colour as for the first simulation in their hand - but this time the dummy bomb has a name on it; also we add a counter with initial value $-1$ to the dummy bomb. Once more replay the log, where this time we also increase  the counter each time a player receives "their" dummy bomb. 
Finally, we reconstruct the original situation: Arrange all players in their starting position, holding the dummy bomb of the same colour as for the first two simulations in their hand, now with a name and a non-negative integer on it. Replace the dummy bombs with armed bombs carrying the same name and counter value.
How many possible logs are there?
If the order of exploders in the same round were important, there would be $\frac{(n+t-1)!}{(t-1)!}$ possible log files (arrange $t$ copies of "bell rings" and $n$ individual "player $i$ explodes", but the first entry must be "bell rings"). However, not every order is compatible with the given original player positions - but certainly each order is possible to achieve by rearranging the initial player positions in one of $n!$ ways. By averaging out over the possible orders, we arrive at $\frac{(n+t-1)!}{(t-1)!n!}={n+t-1\choose n}={n+t-1\choose t-1}$ ways for any given initial arrangement.
So for example, with $t=1$ we find ${n\choose 0}=1$ ways, namely, everybody must start with their left neighbour's bomb and counter set to zero.
Specifically, for $n=8$ and $t=30$, we find
$$ {37\choose 8}=38608020$$
possible starting positions.
