排列球以使相邻球为不同类型的方式

2021年4月4日19:04:10 发表评论 912 次浏览

本文概述

有P型的" p"球, Q型的" q"球和R型的" r"球。使用这些球, 我们要创建一条直线, 以使没有两个相同类型的球相邻。

例子 ：

Input  : p = 1, q = 1, r = 0
Output : 2
There are only two arrangements PQ and QP

Input  : p = 1, q = 1, r = 1
Output : 6
There are only six arrangements PQR, QPR, QRP, RQP, PRQ and RPQ

Input  : p = 2, q = 1, r = 1
Output : 6
There are only six arrangements PQRP, QPRP, PRQP, RPQP, PRPQ and PQPR

天真的解决方案：

这个问题的天真的解决方案是递归解决方案。我们递归地调用三种情况

1)最后要放置的球是P型

2)最后要放置的球是Q型

3)最后要放置的球是R型

以下是上述想法的实现。

C ++

// C++ program to count number of ways to arrange three
// types of balls such that no two balls of same color
// are adjacent to each other
#include<bits/stdc++.h>
using namespace std;
  
// Returns count of arrangements where last placed ball is
// 'last'.  'last' is 0 for 'p', 1 for 'q' and 2 for 'r'
int countWays( int p, int q, int r, int last)
{
     // if number of balls of any color becomes less
     // than 0 the number of ways arrangements is 0.
     if (p<0 || q<0 || r<0)
         return 0;
  
     // If last ball required is of type P and the number
     // of balls of P type is 1 while number of balls of
     // other color is 0 the number of ways is 1.
     if (p==1 && q==0 && r==0 && last==0)
         return 1;
  
     // Same case as above for 'q' and 'r'
     if (p==0 && q==1 && r==0 && last==1)
         return 1;
     if (p==0 && q==0 && r==1 && last==2)
         return 1;
  
     // if last ball required is P and the number of ways is
     // the sum of number of ways to form sequence with 'p-1' P
     // balls, q Q Balls and r R balls ending with Q and R.
     if (last==0)
         return countWays(p-1, q, r, 1) + countWays(p-1, q, r, 2);
  
     // Same as above case for 'q' and 'r'
     if (last==1)
         return countWays(p, q-1, r, 0) + countWays(p, q-1, r, 2);
     if (last==2)
         return countWays(p, q, r-1, 0) + countWays(p, q, r-1, 1);
}
  
// Returns count of required arrangements
int countUtil( int p, int q, int r)
{
    // Three cases arise:
    return countWays(p, q, r, 0) +  // Last required balls is type P
           countWays(p, q, r, 1) +  // Last required balls is type Q
           countWays(p, q, r, 2); // Last required balls is type R
}
  
// Driver code to test above
int main()
{
     int p = 1, q = 1, r = 1;
     printf ( "%d" , countUtil(p, q, r));
     return 0;
}

Java

// Java program to count number 
// of ways to arrange three types of
// balls such that no two balls of 
// same color are adjacent to each other
class GFG {
      
     // Returns count of arrangements
     // where last placed ball is
     // 'last'. 'last' is 0 for 'p', // 1 for 'q' and 2 for 'r'
     static int countWays( int p, int q, int r, int last) 
     {
         // if number of balls of any 
         // color becomes less than 0 
         // the number of ways arrangements is 0.
         if (p < 0 || q < 0 || r < 0 )
         return 0 ;
      
         // If last ball required is
         // of type P and the number
         // of balls of P type is 1
         // while number of balls of
         // other color is 0 the number
         // of ways is 1.
         if (p == 1 && q == 0 && r == 0 && last == 0 )
             return 1 ;
      
         // Same case as above for 'q' and 'r'
         if (p == 0 && q == 1 && r == 0 && last == 1 )
             return 1 ;
         if (p == 0 && q == 0 && r == 1 && last == 2 )
             return 1 ;
      
         // if last ball required is P
         // and the number of ways is
         // the sum of number of ways 
         // to form sequence with 'p-1' P
         // balls, q Q Balls and r R balls
         // ending with Q and R.
         if (last == 0 )
         return countWays(p - 1 , q, r, 1 ) +
                countWays(p - 1 , q, r, 2 );
      
         // Same as above case for 'q' and 'r'
         if (last == 1 )
             return countWays(p, q - 1 , r, 0 ) +
                    countWays(p, q - 1 , r, 2 );
          
         if (last == 2 )
         return countWays(p, q, r - 1 , 0 ) +
                countWays(p, q, r - 1 , 1 );
      
         return 0 ;
     }
      
     // Returns count of required arrangements
     static int countUtil( int p, int q, int r) {
         // Three cases arise:
         return countWays(p, q, r, 0 ) + // Last required balls is type P
                countWays(p, q, r, 1 ) + // Last required balls is type Q
                countWays(p, q, r, 2 );  // Last required balls is type R
     }
      
     // Driver code
     public static void main(String[] args) 
     {
         int p = 1 , q = 1 , r = 1 ;
         System.out.print(countUtil(p, q, r));
     }
}
  
// This code is contributed by Anant Agarwal.

Python3

# Python3 program to count 
# number of ways to arrange 
# three types of balls such  
# that no two balls of same 
# color are adjacent to each 
# other
  
# Returns count of arrangements
# where last placed ball is
# 'last'. 'last' is 0 for 'p', # 1 for 'q' and 2 for 'r'
def countWays(p, q, r, last):
      
     # if number of balls of 
     # any color becomes less
     # than 0 the number of 
     # ways arrangements is 0.
     if (p < 0 or q < 0 or r < 0 ):
         return 0 ;
  
     # If last ball required is
     # of type P and the number
     # of balls of P type is 1
     # while number of balls of
     # other color is 0 the number
     # of ways is 1.
     if (p = = 1 and q = = 0 and
         r = = 0 and last = = 0 ):
         return 1 ;
  
     # Same case as above 
     # for 'q' and 'r'
     if (p = = 0 and q = = 1 and
         r = = 0 and last = = 1 ):
         return 1 ;
          
     if (p = = 0 and q = = 0 and 
         r = = 1 and last = = 2 ):
         return 1 ;
  
     # if last ball required is P 
     # and the number of ways is
     # the sum of number of ways 
     # to form sequence with 'p-1' P
     # balls, q Q Balls and r R 
     # balls ending with Q and R.
     if (last = = 0 ):
         return (countWays(p - 1 , q, r, 1 ) + 
                 countWays(p - 1 , q, r, 2 ));
  
     # Same as above case
     # for 'q' and 'r'
     if (last = = 1 ):
         return (countWays(p, q - 1 , r, 0 ) + 
                 countWays(p, q - 1 , r, 2 ));
     if (last = = 2 ):
         return (countWays(p, q, r - 1 , 0 ) + 
                 countWays(p, q, r - 1 , 1 ));
  
# Returns count of 
# required arrangements
def countUtil(p, q, r):
      
     # Three cases arise:
     # Last required balls is type P
     # Last required balls is type Q
     # Last required balls is type R
     return (countWays(p, q, r, 0 ) + 
             countWays(p, q, r, 1 ) + 
             countWays(p, q, r, 2 )); 
  
# Driver Code
p = 1 ; 
q = 1 ;
r = 1 ;
print (countUtil(p, q, r));
      
# This code is contributed by mits

C#

// C# program to count number 
// of ways to arrange three types of
// balls such that no two balls of 
// same color are adjacent to each other
using System;
  
class GFG {
      
     // Returns count of arrangements
     // where last placed ball is
     // 'last'. 'last' is 0 for 'p', // 1 for 'q' and 2 for 'r'
     static int countWays( int p, int q, int r, int last) 
     {
          
         // if number of balls of any 
         // color becomes less than 0 
         // the number of ways
         // arrangements is 0.
         if (p < 0 || q < 0 || r < 0)
             return 0;
      
         // If last ball required is
         // of type P and the number
         // of balls of P type is 1
         // while number of balls of
         // other color is 0 the number
         // of ways is 1.
         if (p == 1 && q == 0 && r == 0 
                               && last == 0)
             return 1;
      
         // Same case as above for 'q' and 'r'
         if (p == 0 && q == 1 && r == 0 
                                && last == 1)
             return 1;
         if (p == 0 && q == 0 && r == 1 
                                && last == 2)
             return 1;
      
         // if last ball required is P
         // and the number of ways is
         // the sum of number of ways 
         // to form sequence with 'p-1' P
         // balls, q Q Balls and r R balls
         // ending with Q and R.
         if (last == 0)
             return countWays(p - 1, q, r, 1) +
                     countWays(p - 1, q, r, 2);
      
         // Same as above case for 'q' and 'r'
         if (last == 1)
             return countWays(p, q - 1, r, 0) +
                 countWays(p, q - 1, r, 2);
          
         if (last == 2)
             return countWays(p, q, r - 1, 0) +
                     countWays(p, q, r - 1, 1);
      
         return 0;
     }
      
     // Returns count of required arrangements
     static int countUtil( int p, int q, int r)
     {
          
         // Three cases arise:
         // 1. Last required balls is type P
         // 2. Last required balls is type Q
         // 3. Last required balls is type R
         return countWays(p, q, r, 0) + 
                countWays(p, q, r, 1) + 
                countWays(p, q, r, 2); 
     }
      
     // Driver code
     public static void Main() 
     {
         int p = 1, q = 1, r = 1;
          
         Console.Write(countUtil(p, q, r));
     }
}
  
// This code is contributed by nitin mittal.

的PHP

<?php
// PHP program to count number
// of ways to arrange three
// types of balls such that 
// no two balls of same color
// are adjacent to each other
  
// Returns count of arrangements
// where last placed ball is
// 'last'. 'last' is 0 for 'p', // 1 for 'q' and 2 for 'r'
function countWays( $p , $q , $r , $last )
{
      
     // if number of balls of 
     // any color becomes less
     // than 0 the number of 
     // ways arrangements is 0.
     if ( $p < 0 || $q 
            < 0 || $r < 0)
         return 0;
  
     // If last ball required is
     // of type P and the number
     // of balls of P type is 1
     // while number of balls of
     // other color is 0 the number
     // of ways is 1.
     if ( $p == 1 && $q == 0 && 
         $r == 0 && $last == 0)
         return 1;
  
     // Same case as above 
     // for 'q' and 'r'
     if ( $p == 0 && $q == 1 && 
         $r == 0 && $last == 1)
         return 1;
          
     if ( $p == 0 && $q == 0 &&
         $r == 1 && $last == 2)
         return 1;
  
     // if last ball required is P 
     // and the number of ways is
     // the sum of number of ways 
     // to form sequence with 'p-1' P
     // balls, q Q Balls and r R 
     // balls ending with Q and R.
     if ( $last == 0)
         return countWays( $p - 1, $q , $r , 1) + 
                countWays( $p - 1, $q , $r , 2);
  
     // Same as above case
     // for 'q' and 'r'
     if ( $last == 1)
         return countWays( $p , $q - 1, $r , 0) + 
                countWays( $p , $q - 1, $r , 2);
     if ( $last == 2)
         return countWays( $p , $q , $r - 1, 0) + 
                countWays( $p , $q , $r - 1, 1);
}
  
// Returns count of 
// required arrangements
function countUtil( $p , $q , $r )
{
      
     // Three cases arise:
     // Last required balls is type P
     // Last required balls is type Q
     // Last required balls is type R
     return countWays( $p , $q , $r , 0) + 
            countWays( $p , $q , $r , 1) + 
            countWays( $p , $q , $r , 2); 
}
  
     // Driver Code
     $p = 1; 
     $q = 1;
     $r = 1;
     echo (countUtil( $p , $q , $r ));
      
// This code is contributed by nitin mittal.
?>

输出如下：

6

该解决方案的时间复杂度是指数级的。

我们可以观察到有很多子问题一次又一次被解决, 因此可以使用动态编程(DP)解决该问题。我们可以轻松地为该问题提供记忆解决方案。

C ++

// C++ program to count number of ways to arrange three
// types of balls such that no two balls of same color
// are adjacent to each other
#include<bits/stdc++.h>
using namespace std;
#define MAX 100
  
// table to store to store results of subproblems
int dp[MAX][MAX][MAX][3];
  
// Returns count of arrangements where last placed ball is
// 'last'.  'last' is 0 for 'p', 1 for 'q' and 2 for 'r'
int countWays( int p, int q, int r, int last)
{
     // if number of balls of any color becomes less
     // than 0 the number of ways arrangements is 0.
     if (p<0 || q<0 || r<0)
         return 0;
  
     // If last ball required is of type P and the number
     // of balls of P type is 1 while number of balls of
     // other color is 0 the number of ways is 1.
     if (p==1 && q==0 && r==0 && last==0)
         return 1;
  
     // Same case as above for 'q' and 'r'
     if (p==0 && q==1 && r==0 && last==1)
         return 1;
     if (p==0 && q==0 && r==1 && last==2)
         return 1;
  
     // If this subproblem is already evaluated
     if (dp
[q][r][last] != -1)
         return dp
[q][r][last];
  
     // if last ball required is P and the number of ways is
     // the sum of number of ways to form sequence with 'p-1' P
     // balls, q Q Balls and r R balls ending with Q and R.
     if (last==0)
        dp
[q][r][last] = countWays(p-1, q, r, 1) + countWays(p-1, q, r, 2);
  
     // Same as above case for 'q' and 'r'
     else if (last==1)
        dp
[q][r][last] = countWays(p, q-1, r, 0) + countWays(p, q-1, r, 2);
     else //(last==2)
        dp
[q][r][last] =  countWays(p, q, r-1, 0) + countWays(p, q, r-1, 1);
  
     return dp
[q][r][last];
}
  
// Returns count of required arrangements
int countUtil( int p, int q, int r)
{
    // Initialize 'dp' array
    memset (dp, -1, sizeof (dp));
  
    // Three cases arise:
    return countWays(p, q, r, 0) +  // Last required balls is type P
           countWays(p, q, r, 1) +  // Last required balls is type Q
           countWays(p, q, r, 2); // Last required balls is type R
}
  
// Driver code to test above
int main()
{
     int p = 1, q = 1, r = 1;
     printf ( "%d" , countUtil(p, q, r));
     return 0;
}

Java

// Java program to count number
// of ways to arrange three
// types of balls such that no
// two balls of same color
// are adjacent to each other
import java.util.Arrays;
  
class GFG 
{
  
     static final int MAX = 100 ;
      
     // table to store to store results of subproblems
     static int dp[][][][] = new int [MAX][MAX][MAX][ 3 ];
      
     // Returns count of arrangements
     // where last placed ball is
     // 'last'. 'last' is 0 for 'p', // 1 for 'q' and 2 for 'r'
     static int countWays( int p, int q, int r, int last)
     {
         // if number of balls of any 
         // color becomes less than 0
         // the number of ways arrangements is 0.
         if (p < 0 || q < 0 || r < 0 )
         return 0 ;
      
         // If last ball required is 
         // of type P and the number
         // of balls of P type is 1 
         // while number of balls of
         // other color is 0 the number 
         // of ways is 1.
         if (p == 1 && q == 0 && r == 0 && last == 0 )
             return 1 ;
      
         // Same case as above for 'q' and 'r'
         if (p == 0 && q == 1 && r == 0 && last == 1 )
             return 1 ;
          
         if (p == 0 && q == 0 && r == 1 && last == 2 )
             return 1 ;
      
         // If this subproblem is already evaluated
         if (dp
[q][r][last] != - 1 )
             return dp
[q][r][last];
      
         // if last ball required is P and 
         // the number of ways is the sum
         // of number of ways to form sequence
         // with 'p-1' P balls, q Q balss and
         // r R balls ending with Q and R.
         if (last == 0 )
         dp
[q][r][last] = countWays(p - 1 , q, r, 1 ) + 
                             countWays(p - 1 , q, r, 2 );
      
         // Same as above case for 'q' and 'r'
         else if (last == 1 )
         dp
[q][r][last] = countWays(p, q - 1 , r, 0 ) + 
                             countWays(p, q - 1 , r, 2 );
         //(last==2)
         else 
         dp
[q][r][last] = countWays(p, q, r - 1 , 0 ) + 
                             countWays(p, q, r - 1 , 1 );
      
         return dp
[q][r][last];
     }
      
     // Returns count of required arrangements
     static int countUtil( int p, int q, int r)
     {
         // Initialize 'dp' array
         for ( int [][][] row : dp)
         {
             for ( int [][] innerRow : row) 
             {
                 for ( int [] innerInnerRow : innerRow)
                 {
                     Arrays.fill(innerInnerRow, - 1 );
                 }
             }
         };
      
         // Three cases arise:
         return countWays(p, q, r, 0 ) + // Last required balls is type P
             countWays(p, q, r, 1 ) +    // Last required balls is type Q
             countWays(p, q, r, 2 );       // Last required balls is type R
     }
  
     // Driver code
     public static void main(String[] args)
     {
         int p = 1 , q = 1 , r = 1 ;
         System.out.print(countUtil(p, q, r));
     }
}
  
// This code is contributed by Anant Agarwal.

C#

// C# program to count number 
// of ways to arrange three 
// types of balls such that no 
// two balls of same color 
// are adjacent to each other 
using System;
  
class GFG 
{ 
  
     static int MAX = 101; 
      
     // table to store to store results of subproblems 
     static int [, , , ] dp = new int [MAX, MAX, MAX, 4]; 
      
     // Returns count of arrangements 
     // where last placed ball is 
     // 'last'. 'last' is 0 for 'p', // 1 for 'q' and 2 for 'r' 
     static int countWays( int p, int q, int r, int last) 
     { 
         // if number of balls of any 
         // color becomes less than 0 
         // the number of ways arrangements is 0. 
         if (p < 0 || q < 0 || r < 0) 
         return 0; 
      
         // If last ball required is 
         // of type P and the number 
         // of balls of P type is 1 
         // while number of balls of 
         // other color is 0 the number 
         // of ways is 1. 
         if (p == 1 && q == 0 && r == 0 && last == 0) 
             return 1; 
      
         // Same case as above for 'q' and 'r' 
         if (p == 0 && q == 1 && r == 0 && last == 1) 
             return 1; 
          
         if (p == 0 && q == 0 && r == 1 && last == 2) 
             return 1; 
      
         // If this subproblem is already evaluated 
         if (dp
 != -1) 
             return dp
; 
      
         // if last ball required is P and 
         // the number of ways is the sum 
         // of number of ways to form sequence 
         // with 'p-1' P balls, q Q balss and 
         // r R balls ending with Q and R. 
         if (last == 0) 
         dp
 = countWays(p - 1, q, r, 1) + 
                             countWays(p - 1, q, r, 2); 
      
         // Same as above case for 'q' and 'r' 
         else if (last == 1) 
         dp
 = countWays(p, q - 1, r, 0) + 
                             countWays(p, q - 1, r, 2); 
         //(last==2) 
         else
         dp
 = countWays(p, q, r - 1, 0) + 
                             countWays(p, q, r - 1, 1); 
      
         return dp
; 
     } 
      
     // Returns count of required arrangements 
     static int countUtil( int p, int q, int r) 
     { 
         // Initialize 'dp' array 
         for ( int i = 0; i < MAX; i++)
         for ( int j = 0; j < MAX; j++)
         for ( int k = 0; k < MAX; k++)
         for ( int l = 0; l < 4; l++)
         dp[i, j, k, l] = -1;
      
         // Three cases arise: 
         return countWays(p, q, r, 0) + // Last required balls is type P 
             countWays(p, q, r, 1) + // Last required balls is type Q 
             countWays(p, q, r, 2); // Last required balls is type R 
     } 
  
     // Driver code 
     static void Main() 
     { 
         int p = 1, q = 1, r = 1; 
         Console.WriteLine(countUtil(p, q, r)); 
     } 
} 
  
// This code is contributed by mits.

Python3

# Python3 program to count number of ways to 
# arrange three types of balls such that no 
# two balls of same color are adjacent to each other
MAX = 100 ; 
  
# table to store to store results of subproblems 
dp = [[[[ - 1 ] * 4 for i in range ( MAX )] 
                  for j in range ( MAX )] 
                  for k in range ( MAX )]; 
  
# Returns count of arrangements where last 
# placed ball is 'last'. 'last' is 0 for 'p', # 1 for 'q' and 2 for 'r' 
def countWays(p, q, r, last):
  
     # if number of balls of any color becomes less 
     # than 0 the number of ways arrangements is 0. 
     if (p < 0 or q < 0 or r < 0 ): 
         return 0 ; 
  
     # If last ball required is of type P and the 
     # number of balls of P type is 1 while number 
     # of balls of other color is 0 the number of
     # ways is 1. 
     if (p = = 1 and q = = 0 and 
         r = = 0 and last = = 0 ): 
         return 1 ; 
  
     # Same case as above for 'q' and 'r' 
     if (p = = 0 and q = = 1 and
         r = = 0 and last = = 1 ): 
         return 1 ; 
     if (p = = 0 and q = = 0 and 
         r = = 1 and last = = 2 ): 
         return 1 ; 
  
     # If this subproblem is already evaluated 
     if (dp
[q][r][last] ! = - 1 ): 
         return dp






[q][r][last]; 
  
     # if last ball required is P and the number 
     # of ways is the sum of number of ways to  
     # form sequence with 'p-1' P balls, q Q Balls
     # and r R balls ending with Q and R. 
     if (last = = 0 ):
         dp
[q][r][last] = (countWays(p - 1 , q, r, 1 ) + 
                              countWays(p - 1 , q, r, 2 )); 
  
     # Same as above case for 'q' and 'r' 
     elif (last = = 1 ):
         dp
[q][r][last] = (countWays(p, q - 1 , r, 0 ) + 
                              countWays(p, q - 1 , r, 2 )); 
     else :
          
         #(last==2)
         dp
[q][r][last] = (countWays(p, q, r - 1 , 0 ) + 
                              countWays(p, q, r - 1 , 1 )); 
  
     return dp
[q][r][last]; 
  
# Returns count of required arrangements 
def countUtil(p, q, r):
      
     # Three cases arise:
     # Last required balls is type P
     # Last required balls is type Q
     # Last required balls is type R
     return (countWays(p, q, r, 0 ) + 
             countWays(p, q, r, 1 ) + 
             countWays(p, q, r, 2 ));
  
# Driver Code
p, q, r = 1 , 1 , 1 ; 
print (countUtil(p, q, r)); 
  
# This code is contributed by mits

的PHP

<?php
// PHP program to count number of ways to 
// arrange three types of balls such that 
// no two balls of same color are adjacent
// to each other
$MAX = 100; 
  
// table to store to store results of subproblems 
$dp = array_fill (0, $MAX , array_fill (0, $MAX , array_fill (0, $MAX , array_fill (0, 3, -1)))); 
  
// Returns count of arrangements where last 
// placed ball is 'last'. 'last' is 0 for 'p', // 1 for 'q' and 2 for 'r' 
function countWays( $p , $q , $r , $last ) 
{
     global $dp ;
      
     // if number of balls of any color becomes less 
     // than 0 the number of ways arrangements is 0. 
     if ( $p < 0 || $q < 0 || $r < 0) 
         return 0; 
  
     // If last ball required is of type P and the
     // number of balls of P type is 1 while number 
     // of balls of other color is 0 the number of 
     // ways is 1. 
     if ( $p == 1 && $q == 0 && $r == 0 && $last == 0) 
         return 1; 
  
     // Same case as above for 'q' and 'r' 
     if ( $p == 0 && $q == 1 && $r == 0 && $last == 1) 
         return 1; 
     if ( $p == 0 && $q == 0 && $r == 1 && $last == 2) 
         return 1; 
  
     // If this subproblem is already evaluated 
     if ( $dp [ $p ][ $q ][ $r ][ $last ] != -1) 
         return $dp [ $p ][ $q ][ $r ][ $last ]; 
  
     // if last ball required is P and the number of
     // ways is the sum of number of ways to form 
     // sequence with 'p-1' P balls, q Q Balls and r 
     // R balls ending with Q and R. 
     if ( $last == 0) 
     $dp [ $p ][ $q ][ $r ][ $last ] = countWays( $p - 1, $q , $r , 1) + 
                              countWays( $p - 1, $q , $r , 2); 
  
     // Same as above case for 'q' and 'r' 
     else if ( $last == 1) 
     $dp [ $p ][ $q ][ $r ][ $last ] = countWays( $p , $q - 1, $r , 0) + 
                              countWays( $p , $q - 1, $r , 2); 
     else //(last==2) 
     $dp [ $p ][ $q ][ $r ][ $last ] = countWays( $p , $q , $r - 1, 0) + 
                              countWays( $p , $q , $r - 1, 1); 
  
     return $dp [ $p ][ $q ][ $r ][ $last ]; 
} 
  
// Returns count of required arrangements 
function countUtil( $p , $q , $r ) 
{ 
  
// Three cases arise: 
return countWays( $p , $q , $r , 0) + // Last required balls is type P 
        countWays( $p , $q , $r , 1) + // Last required balls is type Q 
        countWays( $p , $q , $r , 2); // Last required balls is type R 
} 
  
// Driver code
$p = 1;
$q = 1;
$r = 1; 
print (countUtil( $p , $q , $r )); 
  
// This code is contributed by mits
?>

输出如下：

6

时间复杂度：O(p * q * r)

辅助空间：O(p * q * r * 3)

如果发现任何不正确的地方, 或者想分享有关上述主题的更多信息, 请写评论。

木子山

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