# 992. Subarrays with K Different Integers

**Difficulty:** Hard

**Topics:** Sliding Window, Hash Table, Two Pointers

**Hint:** Consider using a sliding window approach to count subarrays.

**Problem Description:**

Given an integer array `nums` and an integer `k`, return the number of *good* subarrays of `nums`.

A *good* array is an array where the number of different integers in that array is exactly `k`.

A subarray is a contiguous part of an array.


**Examples:**

**Example 1:**
   
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Input: nums = [1,2,1,2,3], k = 2
Output: 7
Explanation: Subarrays formed with exactly 2 different integers: [1,2], [2,1], [1,2], [2,3], [1,2,1], [2,1,2], [1,2,1,2]
     **Example 2:**
   
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Input: nums = [1,2,1,3,4], k = 3
Output: 3
Explanation: Subarrays formed with exactly 3 different integers: [1,2,1,3], [2,1,3], [1,3,4].
     **Constraints:**

* `1 <= nums.length <= 2 * 10^4`
* `1 <= nums[i], k <= nums.length`
   
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Q- Do we have a custom definition here? 

And array is good when the number of its distinct elements is exactly equal to k 

Q- can you explain the problem using your own words? 

So we are given a list of integer and value k 

→ our goal to find out the number of all the good array in the list  

Q- what is the input& output format ? 

Input : List[int] 
Output : int 

Q- can you go through an example ? 

Input: nums = [1,2,1,2,3], k = 2

Q- always start from naive solution and iteratively improve oky 

The first step is find out all the possible sub_array of different length 

Length 1 [1] , [2] , [1] , [2] ,[3] 

“Please complete for all the length here
Certainly. Let's go through the problem and example step by step, as you've requested.

You're correct about the definition of a "good" array. An array is considered good when the number of its distinct elements is exactly equal to k.

Problem Explanation:
We are given a list of integers (nums) and a value k. Our goal is to find the total number of "good" subarrays in the list. A subarray is any contiguous part of the array, and it's "good" if it contains exactly k distinct integers.

Input & Output Format:
Input: List[int] nums, int k
Output: int (the count of good subarrays)

Let's go through the example you provided:
Input: nums = [1,2,1,2,3], k = 2

First, let's find all possible subarrays of different lengths:

Length 1: [1], [2], [1], [2], [3]
Length 2: [1,2], [2,1], [1,2], [2,3]
Length 3: [1,2,1], [2,1,2], [1,2,3]
Length 4: [1,2,1,2], [2,1,2,3]
Length 5: [1,2,1,2,3]

Now, let's count the "good" subarrays (those with exactly 2 distinct integers):

[1,2], [2,1], [1,2], [2,3], [1,2,1], [2,1,2], [1,2,1,2]

The total count is 7, which is the expected output.

Naive Solution:
A naive approach would be to generate all possible subarrays and count the ones that are "good". Here's a high-level description of this approach:

1. Generate all possible subarrays
2. For each subarray:
   a. Count the number of distinct elements
   b. If the count equals k, increment our result counter
3. Return the final count

This solution would have a time complexity of O(n^3) in the worst case, where n is the length of the input array. This is because we have O(n^2) subarrays, and for each subarray, we might need to count distinct elements, which could take up to O(n) time.

This naive solution would work, but it's not efficient for larger inputs. The next step would be to think about how we can optimize this solution, possibly using the sliding window technique mentioned in the problem tags.

Would you like me to proceed with implementing this naive solution, or shall we move on to discussing a more optimized approach?


Q- now let’s try to opitmaze 

Using dynamaque slide window here 

 Where we scan the array by sliding the window and each window wa apply a afunction transformation 
The function transformation in our example is “set” which compute the number of disitinces element only 
And then comprare with the target value and update the first initiate varileb based on it 
We can use two pointer left and right to implement the dynmique part of the solution oky 
Q- give your feasbakc and formulate my thinking process in mathematical notations oky