To solve Pascal's Triangle, initialize a result array and iteratively build each row using the sum of the two numbers directly above. Start with [1] and append rows using the state transition rule, carefully handling array boundaries. This approach leverages simple dynamic programming and avoids recomputation by building each row from the previous one, making it highly efficient and predictable for interview scenarios.
Problem Statement
Given an integer numRows, generate the first numRows of Pascal's Triangle. Each row should be represented as a list of integers where the first and last elements are 1, and every other element is the sum of the two numbers directly above it from the previous row.
Return the result as a list of lists, where each inner list represents a row of Pascal's Triangle. For example, for numRows = 5, the output should be [[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1]]. Ensure your implementation handles the boundary conditions correctly and constructs the triangle iteratively using dynamic programming.
Examples
Example 1
Input: numRows = 5
Output: [[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1]]
Example details omitted.
Example 2
Input: numRows = 1
Output: [[1]]
Example details omitted.
Constraints
- 1 <= numRows <= 30
Solution Approach
Iterative Dynamic Programming
Start with a result array containing the first row [1]. For each subsequent row, initialize a new list with 1 at the beginning and end. Fill the interior elements using the sum of the two elements directly above in the previous row. Append the completed row to the result. Repeat until numRows are generated. This leverages state transition dynamic programming to build each row efficiently.
Handling Edge Cases
For numRows = 1, immediately return [[1]]. Ensure that the first and last elements of every row remain 1. Dynamic programming logic must account for these boundaries to avoid index errors or incorrect sums, which are common mistakes in Pascal's Triangle problems.
Space Optimization
Although storing all rows is typical, for interview scenarios, one could optimize by keeping only the previous row to generate the current row. This reduces space usage from O(n^2) to O(n), while the core state transition dynamic programming principle remains intact.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(numRows^2) because each row i contains i elements, and we compute each sum once. Space complexity is O(numRows^2) if storing all rows, or O(numRows) if only keeping the previous row for computation.
What Interviewers Usually Probe
- Asks if you can explain why each element is the sum of the two above.
- Wants a clear iterative or dynamic programming solution, not recursion.
- May probe how you handle boundaries or optimize space for large numRows.
Common Pitfalls or Variants
Common pitfalls
- Forgetting to place 1 at the start and end of each row.
- Incorrectly summing elements causing off-by-one errors in the triangle.
- Attempting recursion without memoization, leading to unnecessary complexity.
Follow-up variants
- Generate only the nth row without constructing the full triangle using optimized DP.
- Return Pascal's Triangle modulo a given number for very large values.
- Compute triangle diagonals or specific positions efficiently using combinatorial formulas.
How GhostInterview Helps
- GhostInterview highlights the correct state transition dynamic programming steps row by row.
- It prevents common boundary mistakes by showing safe row construction in arrays.
- Provides stepwise validation for each numRows value to ensure the triangle is built correctly.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the core pattern used in Pascal's Triangle problem?
The main pattern is state transition dynamic programming, where each row is derived from the previous by summing adjacent elements.
Can Pascal's Triangle be generated recursively?
Yes, but recursion without memoization is inefficient and can lead to repeated computations; iterative DP is preferred.
How do I handle the first and last elements in each row?
Always set the first and last elements to 1 explicitly; only sum elements in between using the previous row.
What is the time complexity of generating numRows rows?
Time complexity is O(numRows^2) since row i has i elements and each element is computed once.
Is it possible to optimize space while generating Pascal's Triangle?
Yes, by keeping only the previous row to compute the current row, space can be reduced to O(numRows).
Need direct help with Pascal's Triangle instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Pascal's Triangle from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Compute the specific row of Pascal's Triangle using efficient state transition dynamic programming with array-based updates.
Open problem page#120 TriangleGiven a triangle, return the minimum path sum from top to bottom, moving to adjacent numbers in the row below.
Open problem page#121 Best Time to Buy and Sell StockMaximize stock profit by identifying the optimal buy and sell days using dynamic programming.
Open problem page