How Many Three-Digit Numbers Are There Such That abc 12?

Understanding the Problem:

We need to determine the number of three-digit numbers abc where the product of their digits (a times b times c 12).

Constraints on Digits

Digit a: the hundreds digit must be between 1 and 9 inclusive. Digits b and c: these are the tens and units digits, must be between 0 and 9 inclusive.

Factorizing 12

The prime factorization of 12 is:

12 22 × 31

We will now find all combinations of digits (a, b, c) where the product is 12, considering the constraints for digits.

Valid Combinations of Digits

Let's look at the possible combinations:

1 × 3 × 4 1 × 2 × 6 2 × 2 × 3 1 × 1 × 12 (not valid since 12 is not a digit)

List of Valid Combinations

For 1 × 3 × 4:

134, 143, 314, 341, 413, 431

There are 6 permutations.

For 1 × 2 × 6:

126, 162, 216, 261, 612, 621

There are 6 permutations.

For 2 × 2 × 3:

223, 232, 322

There are 3 permutations.

Calculating the Total Number of Permutations

Adding the valid permutations:

6 from 1 × 3 × 4 6 from 1 × 2 × 6 3 from 2 × 2 × 3

The sum is: 6 6 3 15

Therefore, there are 15 total three-digit numbers (abc) where (a times b times c 12).

Conclusion

In total, there are 15 such three-digit numbers.