Annie can write down any combination of 9 digits where each digit is one of the numbers from 1 to 5. Since the numbers can be repeated, this is a combinatorial problem where we consider all possible ways to arrange the digits 1 through 5 in a 9-digit sequence.

The total number of possible combinations can be calculated as:

$5^9 = 1,953,125$

This represents the number of combinations where each of the 9 positions can be filled by any one of 5 numbers (1 through 5), allowing repetition.

Listing all 1,953,125 combinations would be infeasible here, but to give you an idea, some of the combinations include:

• 111111111
• 111111112
• 111111113
• ...
• 555555555

The key idea is that each of the 9 positions has 5 choices, and all possible choices across the 9 positions create the full set of combinations.

Would you like a further breakdown of how to approach similar problems, or details on combinatorial methods?

Follow-up questions:

1. How would the number of combinations change if Annie only wrote down 5 digits instead of 9?
2. What if Annie could only use distinct numbers instead of allowing repetition?
3. What is the probability of Annie writing the combination "123456789" if she's picking random digits from 1 to 5?
4. Can you calculate how many combinations contain exactly 4 occurrences of the number 5?
5. How would the problem change if Annie selected numbers from 1 to 6 instead of 1 to 5?

Tip: When solving combinatorics problems, first assess whether repetition is allowed and whether order matters (permutations vs. combinations).