1. Introduction

"Pascal's Triangle" is a well-known mathematical concept and a common problem in algorithm design. It involves constructing a triangle of numbers where each number is the sum of the two numbers directly above it. This problem is a great example of using simple iterative algorithms to build complex structures and is often encountered in coding interviews.

Problem

Given an integer numRows, the task is to return the first numRows of Pascal's triangle. Pascal's triangle is a triangular array of integers, where the value at the nth row and kth column is equal to the combination of n and k (n choose k).

2. Solution Steps

1. Initialize a list to hold each row of Pascal's triangle.

2. For each row, initialize a new list with 1 as its first element.

3. Fill in the values of each row based on the sum of the appropriate elements from the previous row.

4. Each element (except the first and last in a row) is the sum of the elements diagonally above it to the left and right.

5. Append the completed row to the triangle list.

6. Repeat steps 2-5 for the specified number of rows.

3. Code Program

def generate(numRows):
    if numRows == 0:
        return []
    triangle = [[1]]

    for i in range(1, numRows):
        row = [1]
        for j in range(1, i):
            row.append(triangle[i - 1][j - 1] + triangle[i - 1][j])
        row.append(1)
        triangle.append(row)

    return triangle

# Example Usage
print(generate(5))

Output:

[
 [1],
 [1, 1],
 [1, 2, 1],
 [1, 3, 3, 1],
 [1, 4, 6, 4, 1]
]

Explanation:

1. Triangle Initialization: A list triangle is initialized to store each row of Pascal's triangle.

2. Iterative Row Construction: Each row is constructed iteratively, starting with 1.

3. Element Calculation: Each element (except the first and last) is calculated by adding the two elements diagonally above it from the previous row.

4. Row Completion: Each row is completed by appending 1 at the end and then added to the triangle.

5. Iterative Building: The process is repeated for the specified number of rows, building up Pascal's triangle.

6. Result: The function returns the first numRows of Pascal's triangle.