算法设计：如何计算二项式系数？（动态规划）

2021年3月27日18:00:40 发表评论 2,528 次浏览

本文概述

以下是的常见定义二项式系数.

二项式系数C(n, k)可定义为(1 + x)^n展开后x^k的系数。
二项式系数C(n, k)也给出了从n个对象中更正式地选择k个对象的方法的数量，而不考虑顺序，即n个元素集合的k个元素子集(或k个组合)的数量。

问题

写一个函数，接受两个参数n和k，并返回二项式系数C(n, k)的值。例如，当n = 4和k = 2时，你的函数应该返回6，当n = 5和k = 2时，它应该返回10。

1)最优子结构

可以使用以下用于二项式系数的标准公式来递归计算C(n, k)的值。

C(n, k) = C(n-1, k-1) + C(n-1, k)
   C(n, 0) = C(n, n) = 1

以下是一个简单的递归实现, 它仅遵循上述递归结构。

C ++

// A naive recursive C++ implementation
#include <bits/stdc++.h>
using namespace std;
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff( int n, int k)
{
     // Base Cases
     if (k == 0 || k == n)
         return 1;
 
     // Recur
     return binomialCoeff(n - 1, k - 1) +
                 binomialCoeff(n - 1, k);
}
 
/* Driver code*/
int main()
{
     int n = 5, k = 2;
     cout << "Value of C(" <<n<< ", " <<k<< ") is " <<
                              binomialCoeff(n, k);
     return 0;
}
 
// This is code is contributed by rathbhupendra

C

// A Naive Recursive Implementation
#include<stdio.h>
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff( int n, int k)
{
   // Base Cases
   if (k==0 || k==n)
     return 1;
 
   // Recur
   return  binomialCoeff(n-1, k-1) +
               binomialCoeff(n-1, k);
}
 
/* Driver program to test above function*/
int main()
{
     int n = 5, k = 2;
     printf ( "Value of C(%d, %d) is %d " , n, k, binomialCoeff(n, k));
     return 0;
}

Java

// JAVA Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
import java.util.*;
 
class GFG {
     
     // Returns value of Binomial
     // Coefficient C(n, k)
     static int binomialCoeff( int n, int k)
     {
     
         // Base Cases
         if (k == 0 || k == n)
             return 1 ;
         
         // Recur
         return binomialCoeff(n - 1 , k - 1 ) +
                     binomialCoeff(n - 1 , k);
     }
     
     /* Driver program to test above function */
     public static void main(String[] args)
    {
         int n = 5 , k = 2 ;
         System.out.printf( "Value of C(%d, %d) is %d " , n, k, binomialCoeff(n, k));
     }
}
 
// This code is contributed by Arnav Kr. Mandal.

python

# A naive recursive Python implementation
 
def binomialCoeff(n , k):
 
     if k = = 0 or k = = n :
         return 1
 
     # Recursive Call
     return binomialCoeff(n - 1 , k - 1 ) +
                 binomialCoeff(n - 1 , k)
 
# Driver Program to test ht above function
n = 5
k = 2
print "Value of C(%d, %d) is (%d)" % (n , k , binomialCoeff(n , k))
 
# This code is contributed by Nikhil Kumar Singh (nickzuck_007)

C#

// C# Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
using System;
 
class GFG {
     
     // Returns value of Binomial
     // Coefficient C(n, k)
     static int binomialCoeff( int n, int k)
     {
         
         // Base Cases
         if (k == 0 || k == n)
             return 1;
         
         // Recur
         return binomialCoeff(n - 1, k - 1) +
                     binomialCoeff(n - 1, k);
     }
     
     /* Driver program to test above function */
     public static void Main()
     {
         int n = 5, k = 2;
         Console.Write( "Value of C(" + n + ", "
                             + k + ") is " +
                         binomialCoeff(n, k));
     }
}
 
// This code is contributed by Sam007.

的PHP

<?php
// PHP Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
 
// Returns value of
// Binomial Coefficient C(n, k)
function binomialCoeff( $n , $k )
{
     // Base Cases
     if ( $k ==0 || $k == $n )
         return 1;
     
     // Recur
     return binomialCoeff( $n - 1, $k - 1) +
                binomialCoeff( $n - 1, $k );
}
 
     // Driver Code
     $n = 5;
     $k = 2;
     echo "Value of C" , "(" , $n , $k , ") is "
                , binomialCoeff( $n , $k );
 
// This code is contributed by aj_36
?>

输出如下

Value of C(5, 2) is 10

2)重叠子问题

应该注意的是, 上述函数一次又一次地计算相同的子问题。有关n = 5和k = 2, 请参见下面的递归树。函数C(3, 1)被调用两次。对于较大的n值, 将存在许多常见的子问题。

二项式系数| DP-91

C(5, 2)的二项式系数递归树

由于再次调用了相同的子问题, 因此此问题具有"重叠子问题"属性。因此, 二项式系数问题具有两个属性(请参阅这个和这个)的动态编程问题。像其他典型的动态编程(DP)问题, 可以通过自下而上的方式构造一个临时2D数组C [] []来避免相同子问题的重新计算。以下是基于动态编程的实现。

C ++

// A Dynamic Programming based solution that uses
// table C[][] to calculate the Binomial Coefficient
#include<bits/stdc++.h>
using namespace std;
 
// Prototype of a utility function that
// returns minimum of two integers
int min( int a, int b);
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff( int n, int k)
{
     int C[n + 1][k + 1];
     int i, j;
 
     // Caculate value of Binomial Coefficient
     // in bottom up manner
     for (i = 0; i <= n; i++)
     {
         for (j = 0; j <= min(i, k); j++)
         {
             // Base Cases
             if (j == 0 || j == i)
                 C[i][j] = 1;
 
             // Calculate value using previously
             // stored values
             else
                 C[i][j] = C[i - 1][j - 1] +
                           C[i - 1][j];
         }
     }
 
     return C[n][k];
}
 
// A utility function to return
// minimum of two integers
int min( int a, int b)
{
     return (a < b) ? a : b;
}
 
// Driver Code
int main()
{
     int n = 5, k = 2;
     cout << "Value of C[" << n << "]["
          << k << "] is " << binomialCoeff(n, k);
}
 
// This code is contributed by Shivi_Aggarwal

C

// A Dynamic Programming based solution
// that uses table C[][] to
// calculate the Binomial Coefficient
#include<stdio.h>
 
// Prototype of a utility function that
// returns minimum of two integers
int min( int a, int b);
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeff( int n, int k)
{
     int C[n+1][k+1];
     int i, j;
 
     // Caculate value of Binomial Coefficient
     // in bottom up manner
     for (i = 0; i <= n; i++)
     {
         for (j = 0; j <= min(i, k); j++)
         {
             // Base Cases
             if (j == 0 || j == i)
                 C[i][j] = 1;
 
             // Calculate value using
             // previously stored values
             else
                 C[i][j] = C[i-1][j-1] + C[i-1][j];
         }
     }
 
     return C[n][k];
}
 
// A utility function to return
// minimum of two integers
int min( int a, int b)
{
     return (a<b)? a: b;
}
 
/* Drier program to test above function*/
int main()
{
     int n = 5, k = 2;
     printf ( "Value of C(%d, %d) is %d " , n, k, binomialCoeff(n, k) );
     return 0;
}

Java

// A Dynamic Programming based
// solution that uses table C[][] to
// calculate the Binomial Coefficient
 
class BinomialCoefficient
{
     // Returns value of Binomial
     // Coefficient C(n, k)
     static int binomialCoeff( int n, int k)
     {
     int C[][] = new int [n+ 1 ][k+ 1 ];
     int i, j;
     
     // Calculate  value of Binomial
     // Coefficient in bottom up manner
     for (i = 0 ; i <= n; i++)
     {
         for (j = 0 ; j <= min(i, k); j++)
         {
             // Base Cases
             if (j == 0 || j == i)
                 C[i][j] = 1 ;
      
             // Calculate value using
             // previously stored values
             else
                 C[i][j] = C[i- 1 ][j- 1 ] + C[i- 1 ][j];
           }
      }
      
     return C[n][k];
     }
 
     // A utility function to return
     // minimum of two integers
     static int min( int a, int b)
     {
     return (a<b)? a: b;
     }
 
     /* Driver program to test above function*/
     public static void main(String args[])
     {
     int n = 5 , k = 2 ;
     System.out.println( "Value of C(" +n+ ", " +k+ ") is " +
                                binomialCoeff(n, k));
     }
}
/*This code is contributed by Rajat Mishra*/

python

# A Dynamic Programming based Python
# Program that uses table C[][]
# to calculate the Binomial Coefficient
 
# Returns value of Binomial Coefficient C(n, k)
def binomialCoef(n, k):
     C = [[ 0 for x in range (k + 1 )] for x in range (n + 1 )]
 
     # Calculate value of Binomial
     # Coefficient in bottom up manner
     for i in range (n + 1 ):
         for j in range ( min (i, k) + 1 ):
             # Base Cases
             if j = = 0 or j = = i:
                 C[i][j] = 1
 
             # Calculate value using
             # previously stored values
             else :
                 C[i][j] = C[i - 1 ][j - 1 ] + C[i - 1 ][j]
 
     return C[n][k]
 
# Driver program to test above function
n = 5
k = 2
print ( "Value of C[" + str (n) + "][" + str (k) + "] is "
       + str (binomialCoef(n, k)))
 
# This code is contributed by Bhavya Jain

C#

// A Dynamic Programming based solution that
// uses table C[][] to calculate the Binomial
// Coefficient
using System;
 
class GFG {
     
     // Returns value of Binomial Coefficient
     // C(n, k)
     static int binomialCoeff( int n, int k)
     {
         int [, ]C = new int [n+1, k+1];
         int i, j;
         
         // Calculate value of Binomial
         // Coefficient in bottom up manner
         for (i = 0; i <= n; i++)
         {
             for (j = 0; j <= Math.Min(i, k); j++)
             {
                 // Base Cases
                 if (j == 0 || j == i)
                     C[i, j] = 1;
         
                 // Calculate value using previously
                 // stored values
                 else
                     C[i, j] = C[i-1, j-1] + C[i-1, j];
             }
         }
         
         return C[n, k];
     }
 
     // A utility function to return minimum
     // of two integers
     static int min( int a, int b)
     {
         return (a < b) ? a : b;
     }
 
     /* Driver program to test above function*/
     public static void Main()
     {
         int n = 5, k = 2;
         Console.WriteLine( "Value of C(" + n
                         + ", " + k + ") is "
                     + binomialCoeff(n, k));
     }
}
 
// This code is contributed by anuj_67.

的PHP

<?php
// A Dynamic Programming based
// solution that uses table C[][] to
// calculate the Binomial Coefficient
 
// Returns value of Binomial
// Coefficient C(n, k)
function binomialCoeff( $n , $k )
{
     $C = array ( array ());
     $i ; $j ;
 
     // Caculate value of Binomial
     // Coefficient in bottom up manner
     for ( $i = 0; $i <= $n ; $i ++)
     {
         for ( $j = 0; $j <= min( $i , $k ); $j ++)
         {
             
             // Base Cases
             if ( $j == 0 || $j == $i )
                 $C [ $i ][ $j ] = 1;
 
             // Calculate value using
             // previously stored values
             else
                 $C [ $i ][ $j ] = $C [ $i - 1][ $j - 1] +
                                  $C [ $i - 1][ $j ];
         }
     }
 
     return $C [ $n ][ $k ];
}
 
     // Driver Code
     $n = 5;
     $k = 2;
     echo "Value of C(" , $n , " " , $k , ") is" , " "
                  , binomialCoeff( $n , $k ) ;
 
// This code is contributed by anuj_67.
?>

输出如下

Value of C[5][2] is 10

时间复杂度：O(n * k)

辅助空间：O(n * k)

下面是上述代码的空间优化版本。下面的代码只使用O(k)。感谢AK提出了这个方法。

C ++

// C++ program for space optimized Dynamic Programming
// Solution of Binomial Coefficient
#include<bits/stdc++.h>
using namespace std;
 
int binomialCoeff( int n, int k)
{
     int C[k+1];
     memset (C, 0, sizeof (C));
 
     C[0] = 1;  // nC0 is 1
 
     for ( int i = 1; i <= n; i++)
     {
         // Compute next row of pascal triangle using
         // the previous row
         for ( int j = min(i, k); j > 0; j--)
             C[j] = C[j] + C[j-1];
     }
     return C[k];
}
 
/* Drier program to test above function*/
int main()
{
     int n = 5, k = 2;
     printf ( "Value of C(%d, %d) is %d " , n, k, binomialCoeff(n, k) );
     return 0;
}

Java

// JAVA Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
import java.util.*;
 
class GFG {
     
     static int binomialCoeff( int n, int k)
     {
         int C[] = new int [k + 1 ];
        
         // nC0 is 1
         C[ 0 ] = 1 ; 
      
         for ( int i = 1 ; i <= n; i++)
         {
             // Compute next row of pascal
             // triangle using the previous row
             for ( int j = Math.min(i, k); j > 0 ; j--)
                 C[j] = C[j] + C[j- 1 ];
         }
         return C[k];
     }
     
     /* Driver program  */
     public static void main(String[] args)
     {
          int n = 5 , k = 2 ;
             System.out.printf( "Value of C(%d, %d) is %d "
                                 , n, k, binomialCoeff(n, k));
     }
}

python

# Python program for Optimized
# Dynamic Programming solution to
# Binomail Coefficient. This one
# uses the concept of pascal
# Triangle and less memory
 
def binomialCoeff(n , k):
 
     # Declaring an empty array
     C = [ 0 for i in xrange (k + 1 )]
     C[ 0 ] = 1 #since nC0 is 1
 
     for i in range ( 1 , n + 1 ):
 
         # Compute next row of pascal triangle using
         # the previous row
         j = min (i , k)
         while (j> 0 ):
             C[j] = C[j] + C[j - 1 ]
             j - = 1
 
     return C[k]
 
# Driver Program to test the above function
n = 5
k = 2
print "Value of C(%d, %d) is %d" % (n, k, binomialCoeff(n, k))
 
# This code is contribtued by Nikhil Kumar Singh(nickzuck_007)

C#

// C# Code for Dynamic Programming |
// Set 9 (Binomial Coefficient)
using System;
 
class GFG {
     
     static int binomialCoeff( int n, int k)
     {
         int [] C = new int [k + 1];
         
         // nC0 is 1
         C[0] = 1;
     
         for ( int i = 1; i <= n; i++)
         {
             // Compute next row of pascal
             // triangle using the previous
             // row
             for ( int j = Math.Min(i, k);
                               j > 0; j--)
                 C[j] = C[j] + C[j-1];
         }
         return C[k];
     }
     
     /* Driver program */
     public static void Main()
     {
         int n = 5, k = 2;
         Console.WriteLine( "Value of C("
                 + n + " " + k + ") is "
                 + binomialCoeff(n, k));
     }
}
 
// This code is contribtued by anuj_67.

的PHP

<?php
// PHP program for space optimized
// Dynamic Programming Solution of
// Binomial Coefficient
function binomialCoeff( $n , $k )
{
     $C = array_fill (0, $k + 1, 0);
 
     $C [0] = 1; // nC0 is 1
 
     for ( $i = 1; $i <= $n ; $i ++)
     {
         // Compute next row of pascal
         // triangle using the previous row
         for ( $j = min( $i , $k ); $j > 0; $j --)
             $C [ $j ] = $C [ $j ] + $C [ $j - 1];
     }
     return $C [ $k ];
}
 
// Driver Code
$n = 5; $k = 2;
echo "Value of C[$n, $k] is " .
         binomialCoeff( $n , $k );
     
// This code is contributed by mits.
?>

输出如下

Value of C(5, 2) is 10

时间复杂度：

O(n * k)

辅助空间：O(k)

说明：

1 ========== >> >> n = 0, C(0, 0)= 1

1–1 ======== >>>> n = 1, C(1, 0)= 1, C(1, 1)= 1

1–2–1 ====== >> n = 2, C(2, 0)= 1, C(2, 1)= 2, C(2, 2)= 1

1–3–3–1 ==== >> n = 3, C(3, 0)= 1, C(3, 1)= 3, C(3, 2)= 3, C(3, 3) = 1

1–4–6–4–1 == >> n = 4, C(4, 0)= 1, C(4, 1)= 4, C(4, 2)= 6, C(4, 3) = 4, C(4, 4)= 1

因此, 这里的每个i循环都使用第(i-1)行建立第i行Pascal三角形

在任何时候, 数组C的每个元素都会有一个值(零或更大), 并且在下一次迭代中, 这些元素的值来自上一次迭代。

在表达式中,

C [j] = C [j] + C [j-1]

右侧代表来自上一迭代的值(Pascal三角形的一行取决于上一行)。左侧代表此语句将获得的当前迭代的值。

Let's say we want to calculate C(4, 3), i.e. n=4, k=3:

All elements of array C of size 4 (k+1) are
initialized to ZERO.

i.e. C[0] = C[1] = C[2] = C[3] = C[4] = 0;
Then C[0] is set to 1

For i = 1:
C[1] = C[1] + C[0] = 0 + 1 = 1 ==>> C(1, 1) = 1

For i = 2:
C[2] = C[2] + C[1] = 0 + 1 = 1 ==>> C(2, 2) = 1
C[1] = C[1] + C[0] = 1 + 1 = 2 ==>> C(2, 1) = 2

For i=3:
C[3] = C[3] + C[2] = 0 + 1 = 1 ==>> C(3, 3) = 1
C[2] = C[2] + C[1] = 1 + 2 = 3 ==>> C(3, 2) = 3
C[1] = C[1] + C[0] = 2 + 1 = 3 ==>> C(3, 1) = 3

For i=4:
C[4] = C[4] + C[3] = 0 + 1 = 1 ==>> C(4, 4) = 1
C[3] = C[3] + C[2] = 1 + 3 = 4 ==>> C(4, 3) = 4
C[2] = C[2] + C[1] = 3 + 3 = 6 ==>> C(4, 2) = 6
C[1] = C[1] + C[0] = 3 + 1 = 4 ==>> C(4, 1) = 4

C(4, 3) = 4 is would be the answer in our example.

记忆方式：这个想法是创建一个查找表并遵循递归自上而下的方法。在计算任何值之前, 我们检查它是否已经在查找表中。如果是, 我们返回值。否则, 我们计算该值并将其存储在查找表中。以下是动态规划的自顶向下方法, 以查找二项式系数的值。

C ++

// A Dynamic Programming based
// solution that uses
// table dp[][] to calculate
// the Binomial Coefficient
// A naive recursive approach
// with table C++ implementation
#include <bits/stdc++.h>
using namespace std;
 
// Returns value of Binomial Coefficient C(n, k)
int binomialCoeffUtil( int n, int k, int ** dp)
{
     // If value in lookup table then return
     if (dp[n][k] != -1) //    
         return dp[n][k];
 
     // store value in a table before return
     if (k == 0) {
         dp[n][k] = 1;
         return dp[n][k];
     }
     
     // store value in table before return
     if (k == n) {
         dp[n][k] = 1;
         return dp[n][k];
     }
     
     // save value in lookup table before return
     dp[n][k] = binomialCoeffUtil(n - 1, k - 1, dp) +
                binomialCoeffUtil(n - 1, k, dp);
     return dp[n][k];
}
 
int binomialCoeff( int n, int k)
{
     int ** dp; // make a temporary lookup table
     dp = new int *[n + 1];
 
     // loop to create table dynamically
     for ( int i = 0; i < (n + 1); i++) {
         dp[i] = new int [k + 1];
     }
 
     // nested loop to initialise the table with -1
     for ( int i = 0; i < (n + 1); i++) {
         for ( int j = 0; j < (k + 1); j++) {
             dp[i][j] = -1;
         }
     }
 
     return binomialCoeffUtil(n, k, dp);
}
 
/* Driver code*/
int main()
{
     int n = 5, k = 2;
     cout << "Value of C(" << n << ", " << k << ") is "
          << binomialCoeff(n, k) << endl;
     return 0;
}
 
// This is code is contributed by MOHAMMAD MUDASSIR

Java

// A Dynamic Programming based
// solution that uses
// table dp[][] to calculate
// the Binomial Coefficient
// A naive recursive approach
// with table Java implementation
import java.util.*;
class GFG{
 
// Returns value of Binomial
// Coefficient C(n, k)
static int binomialCoeffUtil( int n, int k, Vector<Integer> []dp)
{
   // If value in lookup table
   // then return
   if (dp[n].get(k) != - 1 )    
     return dp[n].get(k);
 
   // store value in a table
   // before return
   if (k == 0 )
   {
     dp[n].add(k, 1 );
     return dp[n].get(k);
   }
 
   // store value in table
   // before return
   if (k == n)
   {
     dp[n].add(k, 1 );
     return dp[n].get(k);
   }
 
   // save value in lookup table
   // before return
   dp[n].add(k, binomialCoeffUtil(n - 1 , k - 1 , dp) +
                binomialCoeffUtil(n - 1 , k, dp));
   return dp[n].get(k);
}
 
static int binomialCoeff( int n, int k)
{
   // Make a temporary lookup table
   Vector<Integer> []dp = new Vector[n+ 1 ];
 
   // Loop to create table dynamically
   for ( int i = 0 ; i < (n + 1 ); i++)
   {
     dp[i] = new Vector<Integer>();
     for ( int j = 0 ; j <= k; j++)
       dp[i].add(- 1 );
   }
   return binomialCoeffUtil(n, k, dp);
}
 
// Driver code
public static void main(String[] args)
{
   int n = 5 , k = 2 ;
   System.out.print( "Value of C(" + n +
                    ", " + k + ") is " +
                    binomialCoeff(n, k) + "\n" );
}
}
 
// This code is contributed by Rajput-Ji

Python3

# A Dynamic Programming based solution
# that uses table dp[][] to calculate
# the Binomial Coefficient. A naive
# recursive approach with table
# Python3 implementation
 
# Returns value of Binomial
# Coefficient C(n, k)
def binomialCoeffUtil(n, k, dp):
     
     # If value in lookup table then return
     if dp[n][k] ! = - 1 :
         return dp[n][k]
 
     # Store value in a table before return
     if k = = 0 :
         dp[n][k] = 1
         return dp[n][k]
     
     # Store value in table before return
     if k = = n:
         dp[n][k] = 1
         return dp[n][k]
     
     # Save value in lookup table before return
     dp[n][k] = (binomialCoeffUtil(n - 1 , k - 1 , dp) +
                 binomialCoeffUtil(n - 1 , k, dp))
                 
     return dp[n][k]
 
def binomialCoeff(n, k):
     
     # Make a temporary lookup table
     dp = [ [ - 1 for y in range (k + 1 ) ]
                 for x in range (n + 1 ) ]
 
     return binomialCoeffUtil(n, k, dp)
 
# Driver code
n = 5
k = 2
 
print ( "Value of C(" + str (n) +
                ", " + str (k) + ") is" , binomialCoeff(n, k))
 
# This is code is contributed by Prateek Gupta

C#

// C# program for the
// above approach
 
// A Dynamic Programming based
// solution that uses
// table [, ]dp to calculate
// the Binomial Coefficient
// A naive recursive approach
// with table C# implementation
using System;
using System.Collections.Generic;
class GFG{
 
// Returns value of Binomial
// Coefficient C(n, k)
static int binomialCoeffUtil( int n, int k, List< int > []dp)
{
   // If value in lookup table
   // then return
   if (dp[n][k] != -1)    
     return dp[n][k];
 
   // store value in a table
   // before return
   if (k == 0)
   {
     dp[n][k] = 1;
     return dp[n][k];
   }
 
   // store value in table
   // before return
   if (k == n)
   {
     dp[n][k] = 1;
     return dp[n][k];
   }
 
   // save value in lookup table
   // before return
   dp[n][k] = binomialCoeffUtil(n - 1, k - 1, dp) +
              binomialCoeffUtil(n - 1, k, dp);
   return dp[n][k];
}
 
static int binomialCoeff( int n, int k)
{
   // Make a temporary lookup table
   List< int > []dp = new List< int >[n + 1];
 
   // Loop to create table dynamically
   for ( int i = 0; i < (n + 1); i++)
   {
     dp[i] = new List< int >();
 
     for ( int j = 0; j <= k; j++)
       dp[i].Add(-1);
   }
   return binomialCoeffUtil(n, k, dp);
}
 
// Driver code
public static void Main(String[] args)
{
   int n = 5, k = 2;
   Console.Write( "Value of C(" + n +
                 ", " + k + ") is " +
                 binomialCoeff(n, k) + "\n" );
}
}
 
// This code is contributed by 29AjayKumar

输出如下

Value of C(5, 2) is 10

参考文献：

http://www.csl.mtu.edu/cs4321/www/Lectures/Lecture%2015%20-%20Dynamic%20Programming%20Binomial%20Coefficients.htm

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