How many $3-$digit numbers are there, for which the product of their digits is more than $2$ but less than $7?$

We need to find 3-digit numbers where the product of the digits is more than 2 but less than 7.

So we need: 2 < product of digits < 7

This means the product can be exactly 3, 4, 5, or 6.

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Let's systematically find which combinations of three digits give us products of 3, 4, 5, or 6.

For product = 3:

The only way to get 3 is: 1 × 1 × 3 = 3

Digits: (1, 1, 3)

For product = 4:

1 × 1 × 4 = 4, so digits: (1, 1, 4)

1 × 2 × 2 = 4, so digits: (1, 2, 2)

For product = 5:

The only way to get 5 is: 1 × 1 × 5 = 5

Digits: (1, 1, 5)

For product = 6:

1 × 1 × 6 = 6, so digits: (1, 1, 6)

1 × 2 × 3 = 6, so digits: (1, 2, 3)

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Now we need to count how many different 3-digit numbers can be formed from each set of digits.

When digits repeat, we use the formula for permutations with repetition:

$\tfrac{n!}{n_1! \times n_2! \times ...}$

where $n$ is total digits and $n_1, n_2, ...$ are the frequencies of each repeated digit.

For digits with 2 identical digits:

(1, 1, 3): Number of arrangements = $\tfrac{3!}{2!} = \tfrac{6}{2} = 3$

Arrangements: 113, 131, 311

(1, 1, 4): Number of arrangements = $\tfrac{3!}{2!} = 3$

Arrangements: 114, 141, 411

(1, 1, 5): Number of arrangements = $\tfrac{3!}{2!} = 3$

Arrangements: 115, 151, 511

(1, 1, 6): Number of arrangements = $\tfrac{3!}{2!} = 3$

Arrangements: 116, 161, 611

(1, 2, 2): Number of arrangements = $\tfrac{3!}{2!} = 3$

Arrangements: 122, 212, 221

For digits with all different digits:

(1, 2, 3): Number of arrangements = $3! = 6$

Arrangements: 123, 132, 213, 231, 312, 321

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From sets with repeated digits: 5 sets × 3 arrangements each = 15

From the set with all different digits: 1 set × 6 arrangements = 6

Total = 15 + 6 = 21

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This systematic approach ensures we don't miss any combinations or double-count:

1. We identified all possible digit combinations that give products between 2 and 7

2. We used permutation formulas to count arrangements correctly

3. We separated cases based on whether digits repeat or not

The answer is 21 three-digit numbers.