Answer: The number of permutations taken 3 at a time would be
9! / (3! * 3! * 3!) = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 / (6 * 6 * 6)
= (9 * 8 * 7 * 5 * 4 * 1) / 6 = 8 * 7 * 5 * 6 = 1680
Example: (1 , 2 , 2 ) = 1, 2, 2 2, 1, 2 and 2, 2, 1 3! / 2! = 3
because the 2's are indistinguishable
Now you can put these into groups of 3 and for no reaction all
of the Z bottles must be in 1 group or else a reaction will occur
The number of combinations taken 3 at a time will be
C (9 , 3) = 9! / (3! * 6!) = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 / (3! * 6)
9 * 8 * 7 / 3! = 12 * 7 = 84 groups taken 3 at a time where all of the Z
bottles are in 1 group.
Then the probability of no reaction is 84 / 1680 = .05
and the probability of a reaction is .95
