Count ways to obtain given sum by repeated throws of a dice

Given an integer N, the task is to find the number of ways to get the sum N by repeatedly throwing a dice.

Examples:

Input: N = 3
Output: 4
Explanation:
The four possible ways to obtain N are:

  • 1 + 1 + 1
  • 1 + 2
  • 2 + 1
  • 3

Input: N = 2
Output: 2
Explanation:
The two possible ways to obtain N are:

  • 1 + 1
  • 2

Recursive Approach: The idea is to iterate for every possible value of dice to get the required sum N. Below are the steps:



  1. Let findWays() be the required answer for sum N.
  2. The only numbers obtained from the throw of dice are [1, 6], each having equal probability in a single throw of dice.
  3. Therefore, for every state, recur for the previous (N – i) states (where 1 ≤ i ≤ 6). Therefore, the recursive relation is as follows:

findWays(N) = findWays(N – 1) + findWays(N – 2) + findWays(N – 3) + findWays(N – 4) + findWays(N – 5) + findWays(N – 6)

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the number of ways
// to get the sum N with throw of dice
int findWays(int N)
{
    // Base Case
    if (N == 0) {
        return 1;
    }
 
    // Stores the count of total
    // number of ways to get sum N
    int cnt = 0;
 
    // Recur for all 6 states
    for (int i = 1; i <= 6; i++) {
 
        if (N - i >= 0) {
 
            cnt = cnt
                  + findWays(N - i);
        }
    }
 
    // Return answer
    return cnt;
}
 
// Driver Code
int main()
{
    int N = 4;
 
    // Function call
    cout << findWays(N);
 
    return 0;
}

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Java

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// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Function to find the number of ways
// to get the sum N with throw of dice
static int findWays(int N)
{
     
    // Base Case
    if (N == 0)
    {
        return 1;
    }
 
    // Stores the count of total
    // number of ways to get sum N
    int cnt = 0;
 
    // Recur for all 6 states
    for(int i = 1; i <= 6; i++)
    {
        if (N - i >= 0)
        {
            cnt = cnt +
                  findWays(N - i);
        }
    }
 
    // Return answer
    return cnt;
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 4;
 
    // Function call
    System.out.print(findWays(N));
}
}
 
// This code is contributed by 29AjayKumar

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Python3

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# Python3 program for the above approach
 
# Function to find the number of ways
# to get the sum N with throw of dice
def findWays(N):
     
    # Base case
    if (N == 0):
        return 1
 
    # Stores the count of total
    # number of ways to get sum N
    cnt = 0
 
    # Recur for all 6 states
    for i in range(1, 7):
        if (N - i >= 0):
            cnt = cnt + findWays(N - i)
 
    # Return answer
    return cnt
     
# Driver Code
if __name__ == '__main__':
     
    N = 4
 
    # Function call
    print(findWays(N))
 
# This code is contributed by mohit kumar 29

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C#

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// C# program for
// the above approach
using System;
class GFG{
 
// Function to find the number of ways
// to get the sum N with throw of dice
static int findWays(int N)
{
  // Base Case
  if (N == 0)
  {
    return 1;
  }
 
  // Stores the count of total
  // number of ways to get sum N
  int cnt = 0;
 
  // Recur for all 6 states
  for(int i = 1; i <= 6; i++)
  {
    if (N - i >= 0)
    {
      cnt = cnt + findWays(N - i);
    }
  }
 
  // Return answer
  return cnt;
}
 
// Driver Code
public static void Main()
{
  int N = 4;
 
  // Function call
  Console.Write(findWays(N));
}
}
 
// This code is contributed by sanjoy_62

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Output: 

8









Time Complexity: O(6N)
Auxiliary Space: O(1)

Dynamic Programming Approach: The above recursive approach needs to be optimized by dealing with the following overlapping subproblems:

Overlapping Subproblems: 
Partial recursion tree for N = 8:

Optimal Substructure: As for every state, recurrence occurs for 6 states, so the recursive definition of dp(N) is the following:

dp[N] = dp[N-1] + dp[N-2] + dp[N-3] + dp[N-4] + dp[N-5] + dp[N-6]

Follow the steps below to solve the problem:

  • Initialize an auxiliary array dp[] of size N + 1 with initial value -1, where dp[i] stores the count of ways of having sum i.
  • The base case while solving this problem is if N is equal to 0 in any state, the result to such a state is 1.
  • If for any state dp[i] is not equal to -1, then this value as this substructure is already beed calculated.

Top-Down Approach: Below is the implementation of the Top-Down approach:

C++

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// C++ Program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the total
// number of ways to have sum N
int findWays(int N, int dp[])
{
    // Base Case
    if (N == 0) {
        return 1;
    }
 
    // Return already stored result
    if (dp[N] != -1) {
 
        return dp[N];
    }
 
    int cnt = 0;
 
    // Recur for all 6 states
    for (int i = 1; i <= 6; i++) {
 
        if (N - i >= 0) {
            cnt = cnt
                  + findWays(N - i, dp);
        }
    }
 
    // Return the result
    return dp[N] = cnt;
}
 
// Driver Code
int main()
{
    // Given sum N
    int N = 4;
 
    // Initialize the dp array
    int dp[N + 1];
 
    memset(dp, -1, sizeof(dp));
 
    // Function Call
    cout << findWays(N, dp);
 
    return 0;
}

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Java

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// Java Program for
// the above approach
import java.util.*;
class GFG{
 
// Function to calculate the total
// number of ways to have sum N
static int findWays(int N, int dp[])
{
    // Base Case
    if (N == 0)
    {
        return 1;
    }
 
    // Return already
    // stored result
    if (dp[N] != -1)
    {
        return dp[N];
    }
 
    int cnt = 0;
 
    // Recur for all 6 states
    for (int i = 1; i <= 6; i++)
    {
        if (N - i >= 0)
        {
            cnt = cnt +
                  findWays(N - i, dp);
        }
    }
 
    // Return the result
    return dp[N] = cnt;
}
 
// Driver Code
public static void main(String[] args)
{
    // Given sum N
    int N = 4;
 
    // Initialize the dp array
    int []dp = new int[N + 1];
 
    for (int i = 0; i < dp.length; i++)
        dp[i] = -1;
 
    // Function Call
    System.out.print(findWays(N, dp));
}
}
 
// This code is contributed by 29AjayKumar

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Python3

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# Python3 Program for the
# above approach
 
# Function to calculate
# the total number of ways
# to have sum N
def findWays(N, dp):
 
    # Base Case
    if (N == 0):
        return 1
    
    # Return already
    # stored result
    if (dp[N] != -1):
        return dp[N]
 
    cnt = 0
 
    # Recur for all 6 states
    for i in range (1, 7):
        if (N - i >= 0):
            cnt = (cnt +
                   findWays(N - i, dp))
 
    # Return the result
    dp[N] = cnt
    return dp[N]
 
# Driver Code
if __name__ == "__main__":
 
    # Given sum N
    N = 4
 
    # Initialize the dp array
    dp = [-1] * (N + 1)
 
    # Function Call
    print(findWays(N, dp))
 
# This code is contributed by Chitranayal

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C#

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// C# Program for
// the above approach
using System;
class GFG{
 
// Function to calculate the total
// number of ways to have sum N
static int findWays(int N, int []dp)
{
  // Base Case
  if (N == 0)
  {
    return 1;
  }
 
  // Return already stored result
  if (dp[N] != -1)
  {
    return dp[N];
  }
 
  int cnt = 0;
 
  // Recur for all 6 states
  for (int i = 1; i <= 6; i++)
  {
    if (N - i >= 0)
    {
      cnt = cnt + findWays(N - i, dp);
    }
  }
 
  // Return the result
  return dp[N] = cnt;
}
 
// Driver Code
public static void Main(String[] args)
{
  // Given sum N
  int N = 4;
 
  // Initialize the dp array
  int []dp = new int[N + 1];
 
  for (int i = 0; i < dp.Length; i++)
    dp[i] = -1;
 
  // Function Call
  Console.Write(findWays(N, dp));
}
}
  
// This code is contributed by Rajput-Ji

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Output: 

8









Time Complexity: O(N)
Auxiliary Space: O(N)

Bottom-Up Approach: Below is the implementation of the Bottom-up Dynamic Programming approach:

C++

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// C++ Program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the total
// number of ways to have sum N
void findWays(int N)
{
    // Initialize dp array
    int dp[N + 1];
 
    dp[0] = 1;
 
    // Iterate over all the possible
    // intermediate values to reach N
    for (int i = 1; i <= N; i++) {
 
        dp[i] = 0;
 
        // Calculate the sum for
        // all 6 faces
        for (int j = 1; j <= 6; j++) {
 
            if (i - j >= 0) {
                dp[i] = dp[i] + dp[i - j];
            }
        }
    }
 
    // Print the total number of ways
    cout << dp[N];
}
 
// Driver Code
int main()
{
    // Given sum N
    int N = 4;
 
    // Function call
    findWays(N);
 
    return 0;
}

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Java

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// Java program for the above approach
import java.util.*;
  
class GFG{
  
// Function to calculate the total
// number of ways to have sum N
static void findWays(int N)
{
     
    // Initialize dp array
    int []dp = new int[N + 1];
  
    dp[0] = 1;
  
    // Iterate over all the possible
    // intermediate values to reach N
    for(int i = 1; i <= N; i++)
    {
        dp[i] = 0;
  
        // Calculate the sum for
        // all 6 faces
        for(int j = 1; j <= 6; j++)
        {
            if (i - j >= 0)
            {
                dp[i] = dp[i] + dp[i - j];
            }
        }
    }
  
    // Print the total number of ways
    System.out.print(dp[N]);
}
  
// Driver Code
public static void main(String[] args)
{
     
    // Given sum N
    int N = 4;
  
    // Function call
    findWays(N);
}
}
  
// This code is contributed by Amit Katiyar

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Python3

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# Python3 program for
# the above approach
 
# Function to calculate the total
# number of ways to have sum N
def findWays(N):
   
    # Initialize dp array
    dp = [0] * (N + 1);
    dp[0] = 1;
 
    # Iterate over all the
    # possible intermediate
    # values to reach N
    for i in range(1, N + 1):
        dp[i] = 0;
 
        # Calculate the sum for
        # all 6 faces
        for j in range(1, 7):
            if (i - j >= 0):
                dp[i] = dp[i] + dp[i - j];
 
    # Print total number of ways
    print(dp[N]);
 
# Driver Code
if __name__ == '__main__':
   
    # Given sum N
    N = 4;
 
    # Function call
    findWays(N);
 
# This code is contributed by 29AjayKumar

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C#

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// C# program for
// the above approach
using System;
class GFG{
  
// Function to calculate the total
// number of ways to have sum N
static void findWays(int N)
{
  // Initialize dp array
  int []dp = new int[N + 1];
 
  dp[0] = 1;
 
  // Iterate over all the possible
  // intermediate values to reach N
  for(int i = 1; i <= N; i++)
  {
    dp[i] = 0;
 
    // Calculate the sum for
    // all 6 faces
    for(int j = 1; j <= 6; j++)
    {
      if (i - j >= 0)
      {
        dp[i] = dp[i] + dp[i - j];
      }
    }
  }
 
  // Print the total number of ways
  Console.Write(dp[N]);
}
  
// Driver Code
public static void Main(String[] args)
{
  // Given sum N
  int N = 4;
 
  // Function call
  findWays(N);
}
}
 
// This code is contributed by 29AjayKumar

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Output: 

8









Time Complexity: O(N)
Auxiliary Space: O(N)

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