each draw is from 5 unique objects.

Of course there are 5x5 ways to uniquely select pairs from the 5.

Treat each ball as a base 5 digit. Your possibilities are all 2 digit base 5 numbers.

I see B3's in your list. You only have B1 and B2.

for (b) you only have 3 unique objects now. So there are 3x3=9 combinations. To list them list all 2 digit base 3 numbers.

the probability is $\dfrac 9 {25}$. 9 possible combinations out of 25.