Pascal's Triangle - Java Solution

1. Introduction

This blog post discusses the implementation of Pascal's Triangle, a classic problem in combinatorics. Pascal's Triangle is a triangular array of binomial coefficients, with each number representing the sum of the two numbers directly above it. This problem is fundamental in understanding mathematical properties related to binomial coefficients and combinations.

Problem

Given an integer numRows, the task is to generate the first numRows of Pascal's triangle. Pascal's triangle is constructed such that each row is an expansion of binomial coefficients, starting with row 0 as [1].

2. Solution Steps

1. Initialize a list of lists to store the rows of the triangle.

2. For each row from 0 to numRows - 1, create a new list to represent the row.

3. Populate each row with appropriate binomial coefficients:

a. The first and last element of each row is always 1.

b. Each interior element is the sum of the two elements above it in the triangle.

4. Add each row to the triangle list.

5. Return the list of lists representing Pascal's triangle.

3. Code Program

import java.util.ArrayList;
import java.util.List;

public class PascalsTriangle {
    public static void main(String[] args) {
        int numRows = 5;
        System.out.println(generate(numRows)); // Test the function
    }

    // Function to generate Pascal's triangle
    public static List<List<Integer>> generate(int numRows) {
        List<List<Integer>> triangle = new ArrayList<>();

        for (int i = 0; i < numRows; i++) {
            List<Integer> row = new ArrayList<>();
            for (int j = 0; j <= i; j++) {
                if (j == 0 || j == i) {
                    row.add(1); // First and last element of each row is 1
                } else {
                    row.add(triangle.get(i - 1).get(j - 1) + triangle.get(i - 1).get(j));
                }
            }
            triangle.add(row);
        }

        return triangle;
    }
}

Output:

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

Explanation:

1. generate: This function creates Pascal's triangle up to the numRowsth row. It uses a list of lists to store each row of the triangle.

2. It iterates through each row and initializes a new list for each row.

3. The function populates each row with the correct binomial coefficients:

a. The first and last elements are set to 1.

b. The interior elements are calculated as the sum of the two numbers directly above it in the previous row.

4. Each row is then added to the triangle list.

5. The function returns the completed list of lists, which represents Pascal's triangle up to the given number of rows.

6. This approach efficiently builds Pascal's triangle row by row, leveraging the properties of binomial coefficients.