Question

A person carries 3 bottles of compound X, 3 bottles of compound Y and 3 bottles of compound Z. While X and Y will react vigorously, Z will prevent the reaction from happening. If the person falls and 3 random bottles react, what is the probability that a vigorous reaction will happen?

I tried solving this question like this:
Possible combinations:
ABB
ABA


3/6 x 3/5 x 2/4 = 18/120
3/6 x 3/5 x 2/4 = 18/120


18/120 x 2 = 3/10


I'm wondering if the order of the bottles matter and if the the number of bottles of Z should be counted (e.g. 3/9 x 3/8 x 2/7)?


Please tell me if I'm wrong and provide the right way to do this...

Answer 1

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