12. Choose 6 positive integers from 1 to 900, such that the product of these 6 consecutive positive integers ends with exactly 4 zeros. How many ways are there to do this?
【Analysis】The product ends with exactly 4 zeros, which means it contains exactly 4 factors of 5 and 4 factors of 2.
In any 6 consecutive positive integers, there are at least 3 even numbers, and among any 3 consecutive even numbers, at least one is a multiple of 4, meaning that in any 6 consecutive positive integers, there are at least 4 factors of 2.
Therefore, we only need to consider the number of factors of 5. Since \(5 \times 5 \times 5 \times 5 = 625\), and 625 is within 900, we can choose numbers around 625 to ensure the product ends with exactly 4 zeros. 625 can be the 2nd, 3rd, 4th, or 5th number in the 6 consecutive numbers, providing exactly 4 factors of 5, and thus the product ends with exactly 4 zeros.
If 625 is the 1st or 6th number, and if 625 is the 1st number, then the 6th number is 630, which contains 5 factors of 5, while 626 contains 1 factor of 2, 628 contains 2 factors of 2, and 630 contains 1 factor of 2, providing exactly 4 factors of 2, which is valid.
If 625 is the 6th number, then the 1st number is 620, which contains 5 factors of 5, while 620 contains 2 factors of 2, 622 contains 1 factor of 2, and 624 contains 3 factors of 2, providing 6 factors of 2, and the product would end with 5 zeros, which is invalid.
Therefore, choosing 625 gives 5 valid choices.
If we do not choose 625, we need to have 4 factors of 5, which means there must be 2 multiples of 5 among the 6 numbers. Therefore, the 1st and 6th numbers must be multiples of 5, with one being a multiple of 125 and the other just a multiple of 5.
Since two consecutive multiples of 5 cannot both be multiples of 25.
The multiples of 125 between 1 and 900 are: \(125, 250, 375, 500, 750, 875\), a total of 6 numbers.
Each of these numbers can be the 1st or 6th number in the 6 consecutive integers, giving 2 choices for each number, a total of 12 choices.
In summary, there are 17 valid choices.