Correct Option (2)

Let the 3-digit number be represented as 100a + 10b + c, where a, b, and c are its digits. The reversed number is 100c + 10b + a.

The given conditions are:

All three digits (a, b, c) are different.
The number (100a + 10b + c) is divisible by 7.
The reversed number (100c + 10b + a) is also divisible by 7.

From the second and third conditions, if N = 100a + 10b + c and N' = 100c + 10b + a are both divisible by 7, then their difference (N - N') must also be divisible by 7.

N - N' = (100a + 10b + c) - (100c + 10b + a) = 99a - 99c = 99(a - c).

For 99(a - c) to be divisible by 7, since 99 is not divisible by 7, (a - c) must be divisible by 7. As per the first condition, digits a and c must be different, so a - c ≠ 0. Therefore, the possible values for (a - c) are 7 or -7.

Case 1: a - c = 7

Considering 'a' and 'c' are digits from 0 to 9, and 'a' cannot be 0 (for a 3-digit number) and 'c' cannot be 0 (for the reversed number to be a 3-digit number), the possible pairs for (a, c) are (8, 1) and (9, 2).
Case 2: a - c = -7 (or c - a = 7)

The possible pairs for (a, c) are (1, 8) and (2, 9).

A number 100a + 10b + c is divisible by 7 if and only if (2a + 3b + c) is divisible by 7 (since 100a + 10b + c = 14 * 7a + 7b + 2a + 3b + c, and 14 * 7a + 7b is divisible by 7). We now test each pair (a, c) to find 'b' such that a, b, c are distinct and (2a + 3b + c) is divisible by 7.

For (a, c) = (8, 1):

2(8) + 3b + 1 = 17 + 3b. For this to be divisible by 7, and b ≠ 8, b ≠ 1, if b = 6, then 17 + 3(6) = 35, which is divisible by 7. The digits are 8, 6, 1. The number is 861. Its reverse is 168. Both are divisible by 7.
For (a, c) = (9, 2):

2(9) + 3b + 2 = 20 + 3b. For this to be divisible by 7, and b ≠ 9, b ≠ 2, if b = 5, then 20 + 3(5) = 35, which is divisible by 7. The digits are 9, 5, 2. The number is 952. Its reverse is 259. Both are divisible by 7.
For (a, c) = (1, 8):

2(1) + 3b + 8 = 10 + 3b. For this to be divisible by 7, and b ≠ 1, b ≠ 8, if b = 6, then 10 + 3(6) = 28, which is divisible by 7. The digits are 1, 6, 8. The number is 168. Its reverse is 861. Both are divisible by 7.
For (a, c) = (2, 9):

2(2) + 3b + 9 = 13 + 3b. For this to be divisible by 7, and b ≠ 2, b ≠ 9, if b = 5, then 13 + 3(5) = 28, which is divisible by 7. The digits are 2, 5, 9. The number is 259. Its reverse is 952. Both are divisible by 7.

The four 3-digit numbers satisfying all the given conditions are 168, 259, 861, and 952.

Thus, there are 4 such numbers.

Incorrect Options:

Options 1, 3, and 4 are incorrect because the detailed analysis demonstrates that exactly four numbers meet all the specified criteria, not two, six, or three.