CSES Solutions - Dice Combinations

Your task is to count the number of ways to construct sum N by throwing a dice one or more times. Each throw produces an outcome between 1 and 6.

Examples:

Input: N = 3
Output: 4
Explanation: There are 4 ways to make sum = 3.

• Using 1 die {3}, sum = 3.
• Using 2 dice {1, 2}, sum = 1 + 2 = 3.
• Using 2 dice {2, 1}, sum = 2 + 1 = 3.
• Using 3 dice {1, 1, 1}, sum = 1 + 1 + 1 = 3.

Input: N = 4
Output: 8
Explanation: There are 8 ways to make sum = 4.

• Using 1 die {4}, sum = 4.
• Using 2 dice {1, 3}, sum = 1 + 3 = 4.
• Using 2 dice {2, 2}, sum = 2 + 2 = 4.
• Using 2 dice {3, 1}, sum = 3 + 1 = 4.
• Using 3 dice {1, 1, 2}, sum = 1 + 1 + 2 = 4.
• Using 3 dice {1, 2, 1}, sum = 1 + 2 + 1 = 4.
• Using 3 dice {2, 1, 1}, sum = 2 + 1 + 1 = 4.
• Using 4 dice {1, 1, 1, 1}, sum = 1 + 1 + 1 + 1 = 4.

Approach: To solve the problem, follow the below idea:

The problem can be solved using Dynamic Programming to find the number of ways to construct a particular sum. Maintain a dp[] array such that dp[i] stores the number of ways to construct sum = i. There are only 6 possible sums when we throw a dice: 1, 2, 3, 4, 5 and 6. So, if we want to find the number of ways to construct sum = S after throwing a dice, there are 6 outcomes:

• The dice outcome was 1: So, the number of ways to construct sum = S, will be equal to the number of ways to construct sum = S - 1.
• The dice outcome was 2: So, the number of ways to construct sum = S, will be equal to the number of ways to construct sum = S - 2.
• The dice outcome was 3: So, the number of ways to construct sum = S, will be equal to the number of ways to construct sum = S - 3.
• The dice outcome was 4: So, the number of ways to construct sum = S, will be equal to the number of ways to construct sum = S - 4.
• The dice outcome was 5: So, the number of ways to construct sum = S, will be equal to the number of ways to construct sum = S - 5.
• The dice outcome was 6: So, the number of ways to construct sum = S, will be equal to the number of ways to construct sum = S - 6.

This means to construct a sum S, the total number of ways will be sum of all ways to construct sum from (S - 6) to (S - 1). Now, we can construct the dp[] array as:

dp[i] = ∑(dp[i - j]), where j ranges from 1 to 6.

Step-by-step algorithm:

• Initialize a dp[] array such that dp[i] stores the number of ways we can construct sum = i.
• Initialize dp[0] = 1 as there is only 1 way to make sum = 0, that is to not throw any die at all.
• Iterate i from 1 to X and calculate number of ways to make sum = i.
• Iterate j over all possible values of the last die to make sum = i and add dp[i – j] to dp[i].
• After all the iterations, return the final answer as dp[X].

Below is the implementation of the algorithm:

#include <bits/stdc++.h>
#define ll long long int
using namespace std;

// Function to find the number of ways to construct the
// sum N
ll solve(ll N)
{
    ll mod = 1e9 + 7;

    // dp[] array such that dp[i] stores the number of ways
    // to construct sum = i
    ll dp[N + 1] = {};

    // Initialize dp[0] = 1 as there is only 1 way to
    // construct sum = 0, that is to not throw any dice at
    // all
    dp[0] = 1;

    // Iterate over all possible sums from 1 to N
    for (int i = 1; i <= N; i++) {
        // Iterate over all the possible values of the last die, that is from 1 to 6
        for (int j = 1; j <= 6 && j <= i; j++) {
            dp[i] = (dp[i] + (dp[i - j])) % mod;
        }
    }
    // Return the number of ways to construct sum = N
    return dp[N];
}
int main()
{
    // Sample Input
    ll N = 4;

    cout << solve(N) << "\n";
}

public class DiceSumWays {
    static final long MOD = 1000000007;

    // Function to find the number of ways to construct the sum N
    static long solve(long N) {
        // dp[] array such that dp[i] stores the number of ways
        // to construct sum = i
        long[] dp = new long[(int) (N + 1)];

        // Initialize dp[0] = 1 as there is only 1 way to
        // construct sum = 0, that is to not throw any dice at all
        dp[0] = 1;

        // Iterate over all possible sums from 1 to N
        for (int i = 1; i <= N; i++) {
            // Iterate over all the possible values of the last die, that is from 1 to 6
            for (int j = 1; j <= 6 && j <= i; j++) {
                dp[i] = (dp[i] + dp[(int) (i - j)]) % MOD;
            }
        }

        // Return the number of ways to construct sum = N
        return dp[(int) N];
    }

    public static void main(String[] args) {
        // Sample Input
        long N = 4;

        System.out.println(solve(N));
    }
}

// This code is contributed by shivamgupta310570

using System;

public class GFG {
    static readonly long MOD = 1000000007;
    // Function to find the number of ways to construct the sum N
    static long Solve(long N) {
        long[] dp = new long[N + 1];
        dp[0] = 1;
        // Iterate over all possible sums from 1 to N
        for (int i = 1; i <= N; i++) {
            // Iterate over all the possible values of the last die, that is from 1 to 6
            for (int j = 1; j <= 6 && j <= i; j++) {
                dp[i] = (dp[i] + dp[i - j]) % MOD;
            }
        }
        return dp[N];
    }
    public static void Main(string[] args) {
        // Sample Input
        long N = 4;
        Console.WriteLine(Solve(N));
    }
}

// Function to find the number of ways to construct the
// sum N
function solve(N) {
    const mod = 1e9 + 7;

    // dp[] array such that dp[i] stores the number of ways
    // to construct sum = i
    let dp = new Array(N + 1).fill(0);

    // Initialize dp[0] = 1 as there is only 1 way to
    // construct sum = 0, that is to not throw any dice at
    // all
    dp[0] = 1;

    // Iterate over all possible sums from 1 to N
    for (let i = 1; i <= N; i++) {
        // Iterate over all the possible values of the last die, that is from 1 to 6
        for (let j = 1; j <= 6 && j <= i; j++) {
            dp[i] = (dp[i] + dp[i - j]) % mod;
        }
    }
    // Return the number of ways to construct sum = N
    return dp[N];
}

// Main function
function main() {
    // Sample Input
    let N = 4;

    console.log(solve(N));
}

// Call the main function
main();

MOD = 10**9 + 7

# Function to find the number of ways to construct the sum N
def solve(N):
    # dp[] array such that dp[i] stores the number of ways
    # to construct sum = i
    dp = [0] * (N + 1)

    # Initialize dp[0] = 1 as there is only 1 way to
    # construct sum = 0, that is to not throw any dice at all
    dp[0] = 1

    # Iterate over all possible sums from 1 to N
    for i in range(1, N + 1):
        # Iterate over all the possible values of the last die, that is from 1 to 6
        for j in range(1, 7):
            if j <= i:
                dp[i] = (dp[i] + dp[i - j]) % MOD

    # Return the number of ways to construct sum = N
    return dp[N]

# Sample Input
N = 4
print(solve(N))

8

Time Complexity: O(N), where N is the sum which we need to construct.
Auxiliary Space: O(N)