To find the probability of picking two black balls and one green ball from the basket, we need to calculate the total number of possible outcomes and the number of favorable outcomes.

Total number of possible outcomes:
When three balls are picked at random, the total number of possible outcomes can be calculated using combinations. We need to choose three balls out of the total 15 balls in the basket. Therefore, the total number of possible outcomes is given by the combination formula:

nCr = n! / (r!(n-r)!)

In this case, n = 15 (total number of balls in the basket) and r = 3 (number of balls to be picked).

Total number of possible outcomes = 15C3 = 15! / (3!(15-3)!) = 15! / (3!12!) = (15*14*13) / (3*2*1) = 455

Number of favorable outcomes:
To have two black balls and one green ball, we need to choose 2 black balls out of 4 available black balls and 1 green ball out of 3 available green balls. Therefore, the number of favorable outcomes is given by the combination formula:

nCr = n! / (r!(n-r)!)

In this case, n = 4 (number of black balls) and r = 2 (number of black balls to be picked).

Number of favorable outcomes = 4C2 = 4! / (2!(4-2)!) = 4! / (2!2!) = (4*3) / (2*1) = 6

Similarly, we need to choose 1 green ball out of 3 available green balls.

Number of favorable outcomes = 3C1 = 3! / (1!(3-1)!) = 3! / (1!2!) = (3*2) / (1*2) = 3

Therefore, the number of favorable outcomes is 6 * 3 = 18.

Probability calculation:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = 18 / 455

Simplifying the fraction, we get:

Probability = 2 / 91

Thus, the probability that two balls are black and one ball is green is 2/91.

The correct answer is option E) 18/455.