The logic behind pascal's triangle

Given an integer numRows, return the first numRows of Pascal's triangle.

In Pascal's triangle, each number is the sum of the two numbers directly above it as shown:

leetcode_Pascal_Triangle

Input: numRows = 5 | Output: [[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1]]

According to the question, it is given the number of rows, which means we have to make the loop run till provided integer.
On noticing the GIF of Pascal's triangle we can see the index of the row has the same size as the array
1. [0] 1st row -> 1
2. [1] 2nd row -> 1, 1
3. [2] 3rd row -> 1, 2, 1
4. [3] 4th row -> 1, 3, 3, 1
Another thing we can notice is the pascal triangle each row's starting and end index has a value of 1.
Now, we can assemble this as running a nested loop which runs for each parent loop index which makes it run till that number, if is parent index is 3 for the 3rd row then the child loop index only runs at the same number of time.
Also, creating a new vector at each beginning of the parent loop, will have a dynamic size for the number of rows and all values will be 1.

vector<int>sample(i+1,1);

The child loop starts from the index [2] or 3rd row because that is where we are adding values of the previous row 2nd [1] i.e [1, 1].

// child loop / nested loop

for(int j = 1; j<i;j++){
      // only runs when i became 2
}

About the parent loop now,

for(int i = 0; i<numRows; i++){ // parent loop

      vector<int>sample(i+1,1);

      for(int j = 1; j<i;j++){ // child loop
            // only runs when i became 2
       }

       res.push_back(sample);
}

On 0, the child will not run and the sample will create a vector of size 1 with a value of [1] and push_back to res.
On 1, the child will not run and the sample will create a vector of size 1 with a value of [1, 1] and push_back to res.
Now the child will run and take 1 round to add the value like this:

sample[j]=res[i-1][j-1]+res[i-1][j];

// On i = 2, res[2-1][1-1] + res[2-1][1]
//                res[1][0] + res [1][1]
//                1 + 1 = 2
// sample[1] = 2, while sample is of size 0 and 1,2,1 is stored

Final Code

class Solution {
public:
    vector<vector<int>> generate(int numRows) {
        vector<vector<int>> res;
        // We can't decide the size initially while pushing the array to it

        for(int i = 0; i<numRows; i++){

            vector<int>sample(i+1,1);

            for(int j = 1; j<i;j++){
                sample[j]=res[i-1][j-1]+res[i-1][j];
            }

            res.push_back(sample);
        }

        return res;
    }
};

Now we understood the code in chunks 🤩.

Question link on leetcode.

Pascal's Triangle [ Logic + Solution ] in bits

Given an integer numRows, return the first numRows of Pascal's triangle.

Final Code