To find the number of 4-digit numbers that are divisible by 6, we need to consider the following conditions:

1. The number must be divisible by 2.
2. The number must be divisible by 3.
3. The number should not have any repeated digits.

Condition 1: Divisibility by 2
A number is divisible by 2 if its last digit is even (2, 4, or 6). Since we have 6 digits (1, 2, 3, 4, 5, and 6), half of them are even digits. Therefore, the last digit can be selected in 3 ways.

Condition 2: Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3. We need to select 3 digits from the remaining 5 digits (1, 2, 3, 4, and 5) such that their sum is divisible by 3.

To find the number of ways to select 3 digits, we can use the combination formula. The number of ways to select r objects from a set of n objects is given by nCr = n! / (r! * (n-r)!).

In this case, we need to find the number of ways to select 3 digits from a set of 5 digits, which can be calculated as 5C3 = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1)) = 10.

Therefore, there are 10 ways to select 3 digits such that their sum is divisible by 3.

Condition 3: No repeated digits
Since the number should not have any repeated digits, we can select the remaining digit in 2 ways.

Total number of 4-digit numbers divisible by 6 = Number of ways to select the last digit * Number of ways to select 3 digits for divisibility by 3 * Number of ways to select the remaining digit without repetition

Total number of 4-digit numbers divisible by 6 = 3 * 10 * 2 = 60

Hence, the correct answer is option C) 60.