We need to find how many integers between $1$ and $10^6$ satisfy the condition that the sum of their digits is equal to $9$.

An integer between $1$ and $10^6$ has up to 6 digits. We will treat this problem as finding how many 6-digit numbers (with leading zeros allowed) have their digits summing to 9.

Approach:

1. Interpret the problem as a combinatorial problem: We need to count how many ways we can assign non-negative digits (from 0 to 9) to six positions such that their sum equals 9.

2. Formulate it as a stars and bars problem: Let the six digits of the number be $d_1, d_2, \dots, d_6$, where $d_1 + d_2 + \dots + d_6 = 9$ and $0 \leq d_i \leq 9$ for each $i$.

   The number of solutions to this equation, where the digits can be any non-negative integer, would be a typical "stars and bars" problem: distributing 9 units (stars) among 6 groups (digits). This is calculated using the binomial coefficient: $\binom{9+6-1}{6-1} = \binom{14}{5}.$ This counts all ways of distributing 9 units among 6 positions, without restriction on the size of each position.

3. Account for the digit constraint: The digits $d_1, d_2, \dots, d_6$ are restricted to being between 0 and 9, so we need to ensure no digit exceeds 9. However, since the total sum is 9, which is small compared to 6 digits where each can take values up to 9, no digit can exceed 9 in this case. Thus, no further adjustment is needed.

Final Calculation:

We compute the binomial coefficient: $\binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002.$

Thus, there are 2002 integers between 1 and $10^6$ whose digits sum to 9.

Would you like more details or have any further questions?

Here are 5 related questions to explore further:

1. How would the solution change if the sum of the digits were 15 instead of 9?
2. What if we considered only 5-digit numbers instead of 6-digit numbers?
3. How many integers between 1 and $10^6$ have their digits summing to a value less than 9?
4. What is the probability that a randomly chosen number between 1 and $10^6$ has its digits summing to 9?
5. How does the stars and bars method work in general for problems involving partitions of numbers?

Tip: When solving combinatorial problems, consider if there are restrictions or boundaries that may limit certain solutions. For instance, digit constraints often arise in number problems.