Certain 3-digit numbers have the following characteristics:

1. All the three digits are different.

2. The number is divisible by 7.

3. The number on reversing the digits is also divisible by 7.

How many such 3-digit numbers are there?

The correct answer is 4.

To find the total number of such 3-digit numbers, let us denote the number as:

$N=100a+10b+c$

where $a$, $b$, and $c$ are distinct single digits, and $a\ne 0$ and $c\ne 0$ (since reversing the number must also produce a valid 3-digit number).

The number obtained by reversing the digits is:

$N\prime =100c+10b+a$

Step 1: Use divisibility by 7

According to the question, both $N$ and $N\prime$ are divisible by 7. Therefore, their difference $(N-N\prime )$ must also be divisible by 7.

$N-N\prime =(100a+10b+c)-(100c+10b+a)$

$N-N\prime =99(a-c)$

Since 99 is not divisible by 7, the term $(a-c)$ must be a multiple of 7.

Step 2: Find possible values of $a$ and $c$

Since $a$ and $c$ are non-zero single digits ($1\le a,c\le 9$) and $a\ne c$, the absolute difference $|a-\mathrm{c<mo>|</mo>}$ can only be 7.

This gives two possible pairs for $(a,c)$ and their reverse counterparts:

1. $(a,c)=(8,1)$ or $(1,8)$

2. $(a,c)=(9,2)$ or $(2,9)$

Step 3: Test each pair to find digit $b$

Case 1: $a=8$ and $c=1$

The number is $N=800+10b+1=801+10b$.

Dividing 801 by 7 leaves a remainder of 3 ($801=7\times 114+3$).

For $N$ to be divisible by 7, $3+10b$ (or $3+3b$) must be divisible by 7.

Testing single-digit values of $b$:

If $b=6$, then $801+60=861$, which is divisible by 7 ($861=7\times 123$).

Since all digits (8, 6, 1) are distinct, 861 and its reverse 168 are valid numbers.

Case 2: $a=9$ and $c=2$

The number is $N=900+10b+2=902+10b$.

Dividing 902 by 7 leaves a remainder of 6 ($902=7\times 128+6$).

For $N$ to be divisible by 7, $6+10b$ (or $6+3b$) must be divisible by 7.

Testing single-digit values of $b$:

If $b=5$, then $902+50=952$, which is divisible by 7 ($952=7\times 136$).

Since all digits (9, 5, 2) are distinct, 952 and its reverse 259 are valid numbers.

Conclusion:

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

Thus, there are 4 such 3-digit numbers in total.