Say there a 3 types of Objects A,B,C which randomly interact in pairs to form new objects following the below rules: $$A + B = AB$$ $$B + C = BC$$ $$C + A = CA$$ $$AB + C = ABC$$ $$BC + A = ABC$$ $$CA + B = ABC$$ Say you start with $n_0$ number of A, B and C each. What is the probability that at the end there are only $n_0$ number of ABC left. All interactions are commutative and irreversible.
Clarification: For eg, say a system starts with 2 A's , 2 B's and 2 C's. $$[A,A,B,B,C,C]$$ Randomly choose 2 elements say 1 A and 1 B and they interact to give AB. Now the system is $$[AB,A,B,C,C]$$Randomly choose 2 again say C and C but they don't combine on interaction. So the system remains identical. $$[AB,A,B,C,C]$$ Again randomly choose 2 elements say 1 A and 1 C and they interact to give CA. So the system is now $$[AB,CA,B,C]$$ and so on till the system has no other combining elements like: $$[AB,CA,B,C]\implies[AB,CA,BC]$$ $$OR$$ $$[AB,CA,B,C]\implies[ABC,CA,B]\implies[ABC,ABC]$$
So in a System with these rules, what is the probability that when I start with $n_0$ number of A, B and C each ie. $$[A...n_0\ times,B...n_0\ times,C...n_0\ times]$$ the final state will be $$[ABC...n_0\ times]$$
