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To get one and only one of the two counters to be white, you can either

1)  get a white counter followed by a non-white counter:  (6/10)·(4/9)  =  24/90

2)  or get a non-white counter followed by a white counter:  (4/10)·(6/9)  =  24/90

 

To find the final probability, add those two probabilities together and reduce your answer.