Final answer:
To determine the count of 4-digit numbers with a product of digits equal to 60, we find all combinations of digits that multiply to 60, ensure they do not start with zero and then calculate the permutations for each combination.
Step-by-step explanation:
To find how many 4-digit numbers have a digit product of 60, we must first determine all possible combinations of four single-digit numbers that multiply to 60. The prime factorization of 60 is 22 × 3 × 5. This means we need to consider these factors in groups of four digits, remembering that a 4-digit number cannot start with zero.
Looking at the prime factorization, we can derive combinations such as (1,2,3,10), (1,4,3,5), etc. However, we must be careful with combinations like (1,2,3,10) because numbers like 0x10, which would be a 3-digit number, are not valid. Thus, we should only consider combinations where all numbers are between 1 and 9 inclusive.
Possible valid combinations are (1,2,3,10), (1,2,5,6), (1,3,4,5) and their permutations. However, we only consider the permutations that result in a 4-digit number with no leading zero. So for (1,2,5,6), we could have 1256, 1265, 2156, 2165, and so on. For each valid combination, there are 4! (24) permutations, but we must divide by the number of times 1 appears in the combination, as rearranging 1 has no impact on the number.
In conclusion, to find the total count of such 4-digit numbers, we need to systematically list all valid combinations, consider the permutations, and count them up, avoiding any permutations with a leading zero.