To solve this problem, we need to count the number of integers between 1 and $10^6$ (which means integers from 1 to 999,999) whose digits sum to 15.

Step 1: Define the Problem

We are looking for 6-digit numbers (including those with leading zeros, e.g., 000123 is considered valid) where the sum of the digits equals 15. Essentially, we are distributing 15 "units" among 6 "positions" (digits) where each digit can take values from 0 to 9.

Step 2: Use Stars and Bars Method

This problem can be modeled as a "stars and bars" problem, where we distribute 15 units among 6 positions with each position restricted to the range [0, 9].

However, because each digit must be between 0 and 9, we need to account for the constraints.

Step 3: Generating Function Approach

To find the exact count, we can use the generating function:

$(x^0 + x^1 + \ldots + x^9)^6$

We need the coefficient of $x^{15}$ in the expansion of this polynomial, which corresponds to the count of ways to sum to 15.

Step 4: Coefficient Calculation

The coefficient of $x^{15}$ can be calculated using generating functions or combinatorial counting, often involving inclusion-exclusion to handle the constraints.

Computation

The exact computation of the coefficient $x^{15}$ involves complex algebraic manipulations or programming methods. I will proceed with computing this value directly.The number of integers between 1 and $10^6$ whose digits sum to 15 is 13,992.

Would you like more details about the solution, or do you have any other questions?

Here are some related questions that might interest you:

1. How would the count change if the sum of digits needed to be 10 instead of 15?
2. What is the number of integers whose digits sum to a different value, say 20?
3. How can the stars and bars method be applied to similar digit sum problems?
4. Can this method be extended to solve problems with other digit constraints, such as a maximum digit value less than 9?
5. How does the inclusion-exclusion principle work in counting problems like this?

Tip: For large-scale counting problems involving constraints, generating functions are a powerful tool that helps efficiently manage complex distributions.