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Unique Paths in a Grid

Certification: Intermediate Level Accuracy: 0% Submissions: 0 Points: 10

Write a JavaScript program to find the number of unique paths from the top-left corner to the bottom-right corner of a grid. You can only move either down or right at any point in time. Given a grid of size m × n, calculate the total number of possible unique paths from position (0,0) to position (m-1, n-1).

Example 1
    
  
  • Input: m = 3, n = 7
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  • Output: 28
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  • Explanation: 
    • Grid size is 3×7, start at (0,0), target is (2,6). 
    • At each cell, you can move right or down. 
    • To reach (2,6), you need exactly 2 down moves and 6 right moves. 
    • Total moves = 8, choose 2 positions for down moves. 
    • Number of ways = C(8,2) = 8!/(2!×6!) = 28
    •
Example 2
    
  
  • Input: m = 3, n = 2
    • 
  
  • Output: 3
    • 
  
  • Explanation: 
    • Grid size is 3×2, start at (0,0), target is (2,1). 
    • Path 1: Right → Down → Down. 
    • Path 2: Down → Right → Down. 
    • Path 3: Down → Down → Right. 
    • Total unique paths = 3
    •
Constraints
    
  
  • 1 ≤ m, n ≤ 100
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  • You can only move right or down
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  • Start position is always (0,0)
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  • End position is always (m-1, n-1)
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  • Time Complexity: O(m × n) for DP approach
    • 
  
  • Space Complexity: O(m × n) for DP table, can be optimized to O(n)
    •
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Hint
Solution

Solution Hints

    
  
  • Use dynamic programming to build up solutions from smaller subproblems
    • 
  
  • Create a 2D DP table where dp[i][j] represents paths to reach cell (i,j)
    • 
  
  • Initialize first row and first column to 1 (only one way to reach them)
    • 
  
  • For each cell, paths = paths from left + paths from above
    • 
  
  • Alternative: Use mathematical combination formula C(m+n-2, m-1)
    • 
  
  • Can optimize space by using only one row instead of full 2D table
    •

Steps to solve by this approach:

 Step 1: Create a 2D DP table of size m × n initialized with zeros

 Step 2: Initialize the first row and first column to 1 (base cases)

 Step 3: Iterate through each cell starting from (1,1)

 Step 4: For each cell, calculate paths as sum of paths from left and above

 Step 5: Continue filling the entire DP table row by row

 Step 6: Return the value at the bottom-right corner dp[m-1][n-1]

 Step 7: Optional optimization: use single array instead of 2D table

Submitted Code :