Final answer:
The number of distinguishable permutations of the digits of the number 287,772, when accounting for the repetition of digits, is 60.
Step-by-step explanation:
To find the number of distinguishable permutations of the digits of the number 287,772, we need to consider the number of each digit present in the number and use the formula for permutations of a multiset. The formula is
n! divided by the product of the factorials of the counts of each unique digit. Specifically, for 287,772, we calculate the permutations as follows:
First, count the number of each digit:
- 2 appears twice,
- 7 appears three times,
- 8 appears once.
The number of permutations is the factorial of the total number of digits divided by the product of the factorials of each digit's count:
6! / (2! × 3! × 1!) = (720)/(2 × 6) = 60.
Therefore, there are 60 distinguishable permutations for the number 287,772.