Total number of events = ¹⁵C₆

Problem:
A carton contains 15 apples, out of which 4 are rotten. If a sample of 6 apples is drawn randomly, then find the probability that the sample contains no rotten apples.

Solution:
To find the probability that the sample contains no rotten apples, we need to find the number of favorable outcomes (samples with no rotten apples) and the total number of possible outcomes (all possible samples of 6 apples).

Step 1: Find the number of favorable outcomes:
Since we want to select a sample of 6 apples with no rotten apples, we can only select from the 11 good apples. Therefore, the number of favorable outcomes is the number of ways to select 6 apples from the 11 good apples.

This can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of items and r is the number of items to be selected.

In this case, n = 11 (number of good apples) and r = 6 (number of apples to be selected).

C(11, 6) = 11! / (6!(11-6)!) = 11! / (6!5!) = (11 * 10 * 9 * 8 * 7 * 6!) / (6! * 5 * 4 * 3 * 2 * 1) = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 462

Therefore, the number of favorable outcomes is 462.

Step 2: Find the total number of possible outcomes:
Since we are selecting a sample of 6 apples from a total of 15 apples, the total number of possible outcomes is the number of ways to select 6 apples from the 15 apples.

Using the same combination formula:
C(15, 6) = 15! / (6!(15-6)!) = 15! / (6!9!) = (15 * 14 * 13 * 12 * 11 * 10 * 9!) / (6! * 9!) = (15 * 14 * 13 * 12 * 11 * 10) / (6! * 5 * 4 * 3 * 2 * 1) = 5005

Therefore, the total number of possible outcomes is 5005.

Step 3: Calculate the probability:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

Therefore, the probability that the sample contains no rotten apples is:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 462 / 5005 = 6/65

Therefore, the correct answer is option C) 6/65.