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Compute the Binomial Coefficient

Certification: Intermediate Level Accuracy: 50% Submissions: 52 Points: 10

Write a Python function to compute the binomial coefficient C(n, k), which is the number of ways to choose k elements from a set of n elements.

Example 1
    
  
  • Input: n = 5, k = 2
    • 
  
  • Output: 10
    • 
  
  • Explanation: 
      
    
    • Step 1: Calculate C(5,2) which gives the number of ways to choose 2 elements from a set of 5.
      • 
    
    • Step 2: Using the formula C(n,k) = n! / (k! * (n-k)!), we have C(5,2) = 5! / (2! * 3!).
      • 
    
    • Step 3: This equals 5! / (2! * 3!) = 120 / (2 * 6) = 120 / 12 = 10.
      • 
    
    • Step 4: So there are 10 ways to choose 2 elements from a set of 5 elements.
      • 
  
    •
Example 2
    
  
  • Input: n = 7, k = 3
    • 
  
  • Output: 35
    • 
  
  • Explanation: 
      
    
    • Step 1: Calculate C(7,3) which gives the number of ways to choose 3 elements from a set of 7.
      • 
    
    • Step 2: Using the formula C(n,k) = n! / (k! * (n-k)!), we have C(7,3) = 7! / (3! * 4!).
      • 
    
    • Step 3: This equals 7! / (3! * 4!) = 5040 / (6 * 24) = 5040 / 144 = 35.
      • 
    
    • Step 4: So there are 35 ways to choose 3 elements from a set of 7 elements.
      • 
  
    •
Constraints
    
  
  • 0 <= k <= n <= 50
    • 
  
  • Time Complexity: O(n * k)
    • 
  
  • Space Complexity: O(n * k)
    •
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Hint
Solution

Solution Hints

    
  
  • Use dynamic programming to compute the binomial coefficient.
    • 
  
  • Initialize a 2D array to store intermediate results.
    • 
  
  • Use the formula C(n, k) = C(n-1, k-1) + C(n-1, k).
    •

Steps to solve by this approach:

 Step 1: Create a 2D array (dp) of size (n+1)×(k+1) to store intermediate results.
 
 Step 2: Initialize the base cases in the dp table:
          - dp[i][0] = 1 (ways to choose 0 items from i items)
          - dp[i][i] = 1 (ways to choose i items from i items)
 
 Step 3: Fill the dp table using the recurrence relation: dp[i][j] = dp[i-1][j-1] + dp[i-1][j]
 
 Step 4: Iterate through values of i from 0 to n
 
 Step 5: For each i, iterate through values of j from 0 to min(i,k)
 
 Step 6: Apply the base cases or recurrence formula to fill each cell
 
 Step 7: Return the final result stored in dp[n][k]

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