Explanatory Answer

Method of solving this CAT Quantitative Ability Question from Number System question

Correct Answer : 21

To find the number of 3-digit numbers for which the product of their digits is more than 2 but less than 7, we can analyze the possible combinations of digits.

Let the digits of the 3-digit number be p, q, and r. The product p × q × r should fall within the range 2 < p × q × r < 7.

We can start by considering prime numbers 3 and 5. Since they are prime, they cannot be split into smaller factors. Thus, if one of the digits is 3, the remaining two digits must be 1. This gives us the following combinations:
1, 1, 3
1, 3, 1
3, 1, 1

Next, we have the number 4, which can be split into the factors 2 × 2. The possible combinations with 4 are:
1, 1, 4
1, 4, 1
4, 1, 1
1, 2, 2
2, 1, 2
2, 2, 1

Finally, the number 6 can be split into the factors 3 × 2. The combinations with 6 are:
1, 1, 6
1, 6, 1
6, 1, 1

Additionally, the combination 1, 2, 3 also yields a product of 6. We can arrange these digits in 3! = 6 different ways:
1, 2, 3
1, 3, 2
2, 1, 3
2, 3, 1
3, 1, 2
3, 2, 1

Therefore, by considering all possible combinations, we find a total of 21 three-digit numbers for which the product of their digits is more than 2 but less than 7.