Veritas prep question.

There are three red balls and two black balls in a jar. What is the probability
that John gets at least one red ball when he picks two balls at random out of
the jar at the same time?

The answer is: 9/10

My approach:

I am considering complementary events of taking no red balls.

I may pick up two balls in 5C2 ways.

and then, first black ball may be picked up in 2C1 ways and the other one in 1C1 ways.

Therefore, (2C1*1C1)/(5C2)=1/5

so, probablity of picking one red ball is: 1-(1/5)=4/5.

Where's my approach flawed?