To find the probability of obtaining a sum of 48 when the biased die is thrown thrice, we need to consider all possible combinations of numbers that add up to 48. Let's break down the problem step by step:

Step 1: Find the possible combinations of numbers that add up to 48.
- Since the maximum number on the die is 32, the sum of three numbers cannot exceed 96.
- To find the combinations, we can start with the largest number (32) and then subtract it from 48 to get the remaining sum.
- Let's consider the cases where the largest number is 32, 16, and 8:

Case 1: Largest number is 32:
- In this case, we need to find all possible combinations of two numbers from the remaining set {2, 4, 8, 16, 32} that add up to 16.
- The possible combinations are: (2, 14), (4, 12), (8, 8)
- However, we need to account for the fact that the die is biased and the probability of getting a particular number is 1/n, where n is the number itself.
- So, the probability of getting (2, 14) is (1/2) * (1/14) = 1/28, the probability of getting (4, 12) is (1/4) * (1/12) = 1/48, and the probability of getting (8, 8) is (1/8) * (1/8) = 1/64.

Case 2: Largest number is 16:
- In this case, we need to find all possible combinations of two numbers from the remaining set {2, 4, 8, 16} that add up to 32.
- The possible combinations are: (2, 30), (4, 28), (8, 24), (16, 16)
- The probabilities are: (1/2) * (1/30) = 1/60, (1/4) * (1/28) = 1/112, (1/8) * (1/24) = 1/192, and (1/16) * (1/16) = 1/256.

Case 3: Largest number is 8:
- In this case, we need to find all possible combinations of two numbers from the remaining set {2, 4, 8} that add up to 40.
- The possible combinations are: (2, 38), (4, 36), (8, 32)
- The probabilities are: (1/2) * (1/38) = 1/76, (1/4) * (1/36) = 1/144, and (1/8) * (1/32) = 1/256.

Step 2: Calculate the total probability.
- Now, we need to add up the probabilities from all the cases to get the total probability of obtaining a sum of 48.
- Total Probability = (1/28) + (1/48) + (1/64) + (1/60) + (1/112) + (1/192) + (1/256) + (1/76) + (