Total Commas Across a Large Integer Range
Company: Google
Role: Software Engineer
Category: Coding & Algorithms
Difficulty: medium
Interview Round: Other
### Problem
Write every integer in the inclusive range `[1,n]` in ordinary decimal notation with a comma separating each group of three digits from the right. Return the total number of commas used in all these representations.
Numbers with fewer than four digits contain no commas. For example, `1,000,000` contains two commas. Every comma counts, including multiple commas in the same number.
```cpp
class Solution {
public:
long long countCommas(long long n);
};
```
### Examples
**Example 1:**
```text
Input: n = 1000000
Output: 999002
Explanation: Integers 1,000 through 999,999 contribute 999000 commas, and 1,000,000 contributes two more. Earlier integers contribute none.
```
**Example 2:**
```text
Input: n = 2000000000
Output: 4998999003
Explanation: Counting every comma in all representations through 2,000,000,000 gives 4,998,999,003. Numbers with multiple commas contribute each of them. The total exceeds a signed 32-bit integer.
```
### Constraints
- `1 <= n <= 1000000000000000`.
### Clarifying Questions
- Are both range endpoints included? Yes.
- Does a number with several commas contribute all of them? Yes.
- Is grouping always by three digits from the right? Yes.
- Are leading zeros inserted? No; use the ordinary representation of each positive integer.
- Must the `long long` input and return types be retained? Yes; the allowed range and total exceed 32-bit limits.
### What a Strong Answer Covers
- The exact formatting rule and counting commas rather than merely comma-containing numbers.
- Inclusive boundaries and representations with several commas.
- Correct behavior for small inputs and the largest endpoint.
- A completeness proof, numeric safety, and resource bounds for the full range.
### Follow-up Questions
- How would requesting every formatted string affect the required output size?
- What information would a different digit-grouping convention require?
```hint Count each separator position separately
A number has its first comma once it reaches 1000, its second once it reaches 1000000, and so on. For each power of 1000 not exceeding n, count the integers that reach that threshold and add one contribution for that separator position. Avoid overflowing when forming the next threshold.
```
Overview: Count every comma used across an integer range up to one quadrillion. Covers multiple separators, large totals, exact boundaries, proof, and C++ code.