Pascal’s Triangle and LeetCode Solution in Java
Pascal’s Triangle is a fascinating mathematical concept that has practical applications in various fields, including combinatorics, probability theory, and algebra. In this blog post, we’ll explore what Pascal’s Triangle is, its significance, and how to solve a related problem on LeetCode using Java. We’ll break down the concept into simple language and provide step-by-step examples to make it easy to understand.
What is Pascal’s Triangle?
Pascal’s Triangle is an infinite triangular array of numbers, often used to explore patterns and solve combinatorial problems. It’s named after the French mathematician Blaise Pascal, who studied its properties in the 17th century. The triangle starts with a single number, 1, at the top. Each subsequent row of numbers is constructed by adding the two numbers directly above it. Here’s what the beginning of Pascal’s Triangle looks like:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1As you can see, each row starts and ends with 1, and the interior numbers are the sum of the two numbers above them. This simple rule generates a wealth of interesting patterns and properties.
Pascal’s Triangle and Binomial Coefficients
One of the most significant applications of Pascal’s Triangle is in calculating binomial coefficients. The binomial coefficient, often denoted as “C(n, k)” or “n choose k,” represents the number of ways to choose “k” elements from a set of “n” distinct elements without regard to the order. It’s calculated using Pascal’s Triangle as follows:
C(n, k) = (n choose k) = (n-1 choose k-1) + (n-1 choose k)The binomial coefficient formula can be easily derived from Pascal’s Triangle. Each element in the triangle represents a binomial coefficient. For example, in the fifth row, “1 4 6 4 1” corresponds to C(4, 0), C(4, 1), C(4, 2), C(4, 3), and C(4, 4), respectively.
Pascal’s Triangle and LeetCode
LeetCode is a popular platform for practicing coding problems. There’s a specific problem on LeetCode that involves Pascal’s Triangle, which we’ll tackle using Java. The problem is titled “Pascal’s Triangle II,” and it asks you to return the “kth” row of Pascal’s Triangle, where “k” is a non-negative integer.
Problem Statement
Given an integer “k,” return the “kth” row of Pascal’s Triangle.
Example
Input: k = 3
Output: [1, 3, 3, 1]
Java Solution
To solve this problem in Java, we can create a function that generates Pascal’s Triangle up to the “kth” row and returns that row. Here’s the step-by-step implementation:
import java.util.ArrayList;
import java.util.List;
public class PascalTriangle {
public static List<Integer> getRow(int rowIndex) {
List<Integer> row = new ArrayList<>();
row.add(1); // The first element is always 1
for (int i = 1; i <= rowIndex; i++) {
List<Integer> newRow = new ArrayList<>();
// The first element of each row is always 1
newRow.add(1);
for (int j = 1; j < i; j++) {
int sum = row.get(j - 1) + row.get(j);
newRow.add(sum);
}
// The last element of each row is always 1
newRow.add(1);
// Update the current row with the new row
row = newRow;
}
return row;
}
public static void main(String[] args) {
int k = 3;
List<Integer> result = getRow(k);
System.out.println("Row " + k + " of Pascal's Triangle: " + result);
}
}Explanation of the Code
- We create a function
getRowthat takes an integerrowIndexas input and returns a list of integers representing the “kth” row of Pascal’s Triangle. - We initialize an empty list called
rowto store the current row. - We add the first element of the row, which is always 1, to the
rowlist. - We iterate from the second row to the “kth” row using a
forloop. - Inside the loop, we create a new list called
newRowto store the elements of the current row being generated. - We add the first element of each row, which is always 1, to
newRow. - We iterate through the elements between the first and last (exclusive) of each row, calculating them based on the previous row and adding them to
newRow. - We add the last element of each row, which is always 1, to
newRow. - Finally, we update the
rowvariable withnewRowto prepare for the next iteration. - After the loop completes, we return the
rowlist, which contains the “kth” row of Pascal’s Triangle. - In the
mainfunction, we callgetRowwithk = 3as an example and print the result.
Output
When you run the code with k = 3, you’ll get the following output:
Row 3 of Pascal's Triangle: [1, 3, 3, 1]This output represents the “3rd” row of Pascal’s Triangle, which matches the example provided earlier.
Complexity Analysis
The time complexity of this solution is O(k^2) because we iterate through each row from 1 to k, and for each row, we perform O(k) operations to calculate its elements. The space complexity is O(k) because we store the current row in a list of size k.
Conclusion
Pascal’s Triangle is a fascinating mathematical concept with practical applications in combinatorics and beyond. Understanding its properties and patterns can help you solve various mathematical and coding problems, as demonstrated by the LeetCode problem “Pascal’s Triangle II.” We’ve provided a step-by-step Java solution to this problem, making use of Pascal’s Triangle’s recursive nature to generate the desired row efficiently.
By mastering concepts like Pascal’s Triangle and practicing coding problems, you can enhance your problem-solving skills and become a more proficient programmer. Whether you’re preparing for coding interviews or simply exploring the beauty of mathematics, Pascal’s Triangle is a valuable topic to explore.
Thank You!
Read More –
Two Sum II – Input Array Is Sorted | Two Sum II – Input Array Is Sorted LeetCode Solution in Java – https://kamleshsingad.com/two-sum-ii-input-array-is-sorted-leetcode-solution-in-java/
234. Palindrome Linked List | Palindromic Linked List LeetCode Solution in Java | Code with Kamlesh – https://kamleshsingad.com/234-palindrome-linked-list-and-leetcode-solution-in-java/
Valid Palindrome and LeetCode Solution in Java: A Detailed Guide – https://kamleshsingad.com/valid-palindrome-and-leetcode-solution-in-java-a-detailed-guide/
