This is my 2 cent worth
There are a number of was to look at this question. If it was part of a probablitiy question you would assume every ball was numbered as well as coloured
BUT this is NOT a probablily question so I am going to interprete it differently
Added after: The title is Permutation which means that Order Does Count
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First interpretation: order does not count. Here are my choices.
BBB, BBR, BBW, BRR, BWW, BRW That makes 6 outcomes
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Now I am going to look at if order does count.
Here are the choices
BBB
BBR or BBW
BRB or BWB
RBB or WBB
Now if I chose just 2 non-black ones then I can have WW,WR,RW, or RR now the black one can be 1st second or third so that is 4*3=12
Total = 1+6+12=19 ways [This is the only one with any validity out of my 3]
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NOW LETS LOOK AT IT FROM AN ORDER DOES NOT COUNT PROBABILITY PERSPECIVE
Total number of ways that you can choose 3 b***s from 9 is 9C3=84
Total number of ways of not choosing any black ones is 6C3 = 20
There for total number of ways of haveing a black one in the mix must be 84-20=64
(This is the same answer as most the others got I think)
Lets do this the long way and see if there is any agreement
all black 3C3=1
2black and 1 other = 3C2*6C1=3*6=18
1black and 2 other = 3C1*6C2=3*15=45
1+18+45=64 GOOD IT IS THE SAME
P(choosing at least 1 black out of 3 where order doesnt count.) = 64/84
64 IS NOT THE CORRECT ANSWER BECAUSE THE TITLE OF THIS QUESTION IS PERMUTATION.
64 IS THE ANSWER TO A COMBINATION QUESTION SO IT IS NOT CORRECT !!
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Now how about if order does count from a proabability perspective
It is all getting a bit much now. 


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