Final answer:
There are 1120 different ways to arrange the digits in the number 9,128,821. This is calculated using the permutation formula for a multiset, taking into account the repetitions of '2' and '8'.
Step-by-step explanation:
To determine the number of ways the digits in the number 9,128,821 can be arranged, we have to consider that some digits are repeated. The number has a total of 8 digits consisting of one '9', one '1', three '2's, and three '8's. To calculate the total number of arrangements, we use the formula for permutations of a multiset, which is the factorial of the total number of digits divided by the product of the factorials of the counts of each unique digit.
The formula for this would be:
Arrangement = \( \frac{8!}{3! \times 3!} \)
Calculating this, it yields:
Arrangement = \( \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) \times (3 \times 2 \times 1)} \) = \( \frac{40320}{36} \) = 1120
Therefore, there are 1120 different ways to arrange the digits in the number 9,128,821.