Add documentation for LeetCode problem 992: Subarrays with K Different Integers

- Created a new Markdown file detailing the problem statement, solution approach, and algorithm for counting subarrays with exactly K distinct integers using a sliding window technique.
- Included a comprehensive summary, time and space complexity analysis, example walkthroughs, and edge cases to enhance understanding.
- Provided implementation details and related problems for further exploration.

Author: HaoweiChan

notes/leetcode/992-subarrays-with-k-different-integers.md

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+---
+title: "992. Subarrays with K Different Integers"
+date: "2025-10-08"
+tags: ["Hash Table", "Two Pointers", "Sliding Window"]
+related: []
+slug: "992-subarrays-with-k-different-integers"
+category: "leetcode"
+leetcode_url: "https://leetcode.com/problems/subarrays-with-k-different-integers/description/?envType=problem-list-v2&envId=xxzz5vc6"
+leetcode_difficulty: "Hard"
+leetcode_topics: ["Hash Table", "Two Pointers", "Sliding Window"]
+---
+
+# 992. Subarrays with K Different Integers
+
+## Summary
+Given an integer array nums and an integer k, return the number of subarrays that contain exactly k different integers using a sliding window approach with the formula: exactly K = at most K - at most (K-1).
+
+## Problem Description
+Given an integer array `nums` and an integer `k`, return the number of subarrays that contain **exactly** `k` distinct integers.
+
+**Constraints:**
+- `1 <= nums.length <= 2 * 10^4`
+- `1 <= nums[i] <= nums.length`
+- `1 <= k <= nums.length`
+
+## Solution Approach
+This problem can be solved using the **sliding window** technique with a clever mathematical insight:
+
+**Key Insight:** The number of subarrays with **exactly K** distinct integers = number of subarrays with **at most K** distinct integers - number of subarrays with **at most (K-1)** distinct integers.
+
+**Algorithm:**
+1. Create a helper function that counts subarrays with **at most** `max_distinct` integers
+2. Use a sliding window with a frequency map to track distinct elements
+3. For each right pointer position, expand the window until distinct count exceeds `max_distinct`
+4. Count valid windows as `right - left + 1` for each valid window
+5. Return `at_most(K) - at_most(K-1)` for exactly K
+
+**Implementation Details:**
+- Use a dictionary to track frequency of each number in current window
+- Use a counter for distinct elements in current window
+- When distinct count exceeds limit, shrink window from left until it's valid again
+- For each valid window ending at `right`, add `right - left + 1` to the count
+
+## Time & Space Complexity
+- **Time Complexity:** O(n) - each element is visited at most twice (once when expanding, once when contracting)
+- **Space Complexity:** O(n) - for the frequency dictionary in worst case
+
+## Key Insights
+- The "exactly K = at most K - at most (K-1)" formula is the key insight that makes this problem solvable in linear time
+- We need to handle the case where K = 1 separately (no K-1 case)
+- The sliding window approach works because we can efficiently maintain the count of distinct elements
+- This technique is generalizable to other "exactly K" problems
+
+## Examples / snippets
+
+### Solution Code
+```python
+from typing import List
+from collections import defaultdict
+
+class Solution:
+    def subarraysWithKDistinct(self, nums: List[int], k: int) -> int:
+        def at_most(k_val: int) -> int:
+            count = 0
+            left = 0
+            freq = defaultdict(int)
+
+            for right in range(len(nums)):
+                freq[nums[right]] += 1
+
+                # Shrink window if too many distinct elements
+                while len(freq) > k_val:
+                    freq[nums[left]] -= 1
+                    if freq[nums[left]] == 0:
+                        del freq[nums[left]]
+                    left += 1
+
+                # Add all valid subarrays ending at 'right'
+                count += right - left + 1
+
+            return count
+
+        # Exactly K = At most K - At most (K-1)
+        return at_most(k) - at_most(k - 1)
+```
+
+### Example Walkthrough
+**Example 1:**
+```
+Input: nums = [1,2,1,2,3], k = 2
+Output: 7
+
+Explanation:
+Subarrays with exactly 2 distinct integers:
+- [1,2] (indices 0-1)
+- [2,1] (indices 1-2)
+- [1,2] (indices 1-3, but wait, let's list properly)
+- [1,2,3] (indices 0-4, but this has 3 distinct)
+- [2,1,2] (indices 1-3, has 2 distinct)
+- [1,2,3] (indices 2-4, has 3 distinct)
+- [2,3] (indices 3-4, has 2 distinct)
+
+Wait, let me be more careful:
+- Length 2: [1,2], [2,1], [1,2], [2,3] = 4
+- Length 3: [1,2,1], [2,1,2] = 2
+- Length 4: [1,2,1,2] = 1
+Total: 7 ✓
+
+Using formula: at_most(2) - at_most(1)
+- at_most(2) = 10 (all subarrays with ≤2 distinct)
+- at_most(1) = 3 ([1], [2], [3] length 1)
+- 10 - 3 = 7 ✓
+```
+
+**Example 2:**
+```
+Input: nums = [1,2,1,3,4], k = 3
+Output: 3
+
+Subarrays: [1,2,1,3], [2,1,3,4], [1,3,4]
+Each has exactly 3 distinct integers.
+```
+
+## Edge Cases & Validation
+- **K = 1**: Should return number of subarrays with exactly 1 distinct integer
+- **K = len(nums)**: Should return 1 (the entire array, if it has exactly K distinct)
+- **All elements same**: `[1,1,1,1], k=1` should return 10 (all subarrays)
+- **All elements different**: `[1,2,3,4], k=4` should return 1
+- **Empty array**: Not possible per constraints (n >= 1)
+- **Single element**: `[5], k=1` should return 1
+
+## Related Problems
+- [340. Longest Substring with At Most K Distinct Characters](https://leetcode.com/problems/longest-substring-with-at-most-k-distinct-characters/) - Similar sliding window
+- [3. Longest Substring Without Repeating Characters](https://leetcode.com/problems/longest-substring-without-repeating-characters/) - K= all distinct
+- [904. Fruit Into Baskets](https://leetcode.com/problems/fruit-into-baskets/) - Exactly 2 types
+- [930. Binary Subarrays With Sum](https://leetcode.com/problems/binary-subarrays-with-sum/) - Similar prefix sum technique
+```