Problem Statement
Row Extractor
You're a mathematics researcher studying combinatorial patterns. You need to extract a specific row from Pascal's Triangle for your calculations, but you want to do it efficiently without generating the entire triangle.
In Pascal's Triangle, each number is the sum of the two numbers directly above it. The first row is always [1].
Given a row index rowIndex (0-indexed), your task is to return the rowIndexth row of Pascal's Triangle.
For example, when rowIndex = 3, you should return the 4th row [1,3,3,1].
Follow-up: Could you optimize your algorithm to use only O(rowIndex) extra space?
Examples
Example 1:
Input: rowIndex = 3
Output: [1,3,3,1]
Explanation: The 4th row (0-indexed 3rd row) of Pascal's Triangle.
Example 2:
Input: rowIndex = 0
Output: [1]
Explanation: The 1st row (0-indexed 0th row) is just [1].
Example 3:
Input: rowIndex = 1
Output: [1,1]
Explanation: The 2nd row (0-indexed 1st row) is [1,1].
Constraints
- 0 <= rowIndex <= 33
Problem Breakdown
To solve this problem, we need to:
- Each row starts and ends with 1
- Each element inside a row is the sum of the two elements above it
- The rowIndex-th row has rowIndex+1 elements
- The row is symmetric around its center
- The sum of the elements in the rowIndex-th row is 2^rowIndex
- Each row represents the coefficients of the binomial expansion (a+b)^rowIndex
- We can compute the row directly using the formula C(rowIndex, i) = rowIndex! / (i! * (rowIndex-i)!)
- We can also compute the row iteratively using the property C(n, k) = C(n, k-1) * (n-k+1) / k
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Row Extractor
You're a mathematics researcher studying combinatorial patterns. You need to extract a specific row from Pascal's Triangle for your calculations, but you want to do it efficiently without generating the entire triangle.
In Pascal's Triangle, each number is the sum of the two numbers directly above it. The first row is always [1].
Given a row index rowIndex (0-indexed), your task is to return the rowIndexth row of Pascal's Triangle.
For example, when rowIndex = 3, you should return the 4th row [1,3,3,1].
Follow-up: Could you optimize your algorithm to use only O(rowIndex) extra space?
Examples
Example 1:
The 4th row (0-indexed 3rd row) of Pascal's Triangle.
Example 2:
The 1st row (0-indexed 0th row) is just [1].
Example 3:
The 2nd row (0-indexed 1st row) is [1,1].
Problem Breakdown
Constraints:
- 0 <= rowIndex <= 33
Key Insights:
Each row starts and ends with 1
Each element inside a row is the sum of the two elements above it
The rowIndex-th row has rowIndex+1 elements
The row is symmetric around its center
The sum of the elements in the rowIndex-th row is 2^rowIndex
Each row represents the coefficients of the binomial expansion (a+b)^rowIndex
We can compute the row directly using the formula C(rowIndex, i) = rowIndex! / (i! * (rowIndex-i)!)
We can also compute the row iteratively using the property C(n, k) = C(n, k-1) * (n-k+1) / k
Real-World Application
This problem has several practical applications:
Mathematics Research
Studying combinatorial patterns and binomial coefficients in advanced mathematics.
Probability Calculations
Computing probabilities in various scenarios, such as coin flips and dice rolls.
Combinatorial Algorithms
Solving problems related to combinations and permutations in computer science.
Statistical Analysis
Analyzing data distributions and calculating binomial distributions.