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Dynamic Programming

3183. The Number of Ways to Make the Sum 🔒

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Description

You have an infinite number of coins with values 1, 2, and 6, and only 2 coins with value 4.

Given an integer n, return the number of ways to make the sum of n with the coins you have.

Since the answer may be very large, return it modulo 109 + 7.

Note that the order of the coins doesn't matter and [2, 2, 3] is the same as [2, 3, 2].

 

Example 1:

Input: n = 4

Output: 4

Explanation:

Here are the four combinations: [1, 1, 1, 1], [1, 1, 2], [2, 2], [4].

Example 2:

Input: n = 12

Output: 22

Explanation:

Note that [4, 4, 4] is not a valid combination since we cannot use 4 three times.

Example 3:

Input: n = 5

Output: 4

Explanation:

Here are the four combinations: [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], [1, 4].

 

Constraints:

  • 1 <= n <= 105

Solutions

Solution 1: Dynamic Programming (Complete Knapsack)

We can start by ignoring coin $4$, defining the coin array coins = [1, 2, 6], and then using the idea of the complete knapsack problem. We define $f[j]$ as the number of ways to make up amount $j$ using the first $i$ types of coins, initially $f[0] = 1$. Then, we iterate through the coin array coins, and for each coin $x$, we iterate through amounts from $x$ to $n$, updating $f[j] = f[j] + f[j - x]$.

Finally, $f[n]$ is the number of ways to make up amount $n$ using coins $1, 2, 6$. Then, if $n \geq 4$, we consider choosing one coin $4$, so the number of ways becomes $f[n] + f[n - 4]$, and if $n \geq 8$, we consider choosing two coins $4$, so the number of ways becomes $f[n] + f[n - 4] + f[n - 8]$.

Note the modulus operation for the answer.

The time complexity is $O(n)$, and the space complexity is $O(n)$. Where $n$ is the amount.

Python3

class Solution:
    def numberOfWays(self, n: int) -> int:
        mod = 10**9 + 7
        coins = [1, 2, 6]
        f = [0] * (n + 1)
        f[0] = 1
        for x in coins:
            for j in range(x, n + 1):
                f[j] = (f[j] + f[j - x]) % mod
        ans = f[n]
        if n >= 4:
            ans = (ans + f[n - 4]) % mod
        if n >= 8:
            ans = (ans + f[n - 8]) % mod
        return ans

Java

class Solution {
    public int numberOfWays(int n) {
        final int mod = (int) 1e9 + 7;
        int[] coins = {1, 2, 6};
        int[] f = new int[n + 1];
        f[0] = 1;
        for (int x : coins) {
            for (int j = x; j <= n; ++j) {
                f[j] = (f[j] + f[j - x]) % mod;
            }
        }
        int ans = f[n];
        if (n >= 4) {
            ans = (ans + f[n - 4]) % mod;
        }
        if (n >= 8) {
            ans = (ans + f[n - 8]) % mod;
        }
        return ans;
    }
}

C++

class Solution {
public:
    int numberOfWays(int n) {
        const int mod = 1e9 + 7;
        int coins[3] = {1, 2, 6};
        int f[n + 1];
        memset(f, 0, sizeof(f));
        f[0] = 1;
        for (int x : coins) {
            for (int j = x; j <= n; ++j) {
                f[j] = (f[j] + f[j - x]) % mod;
            }
        }
        int ans = f[n];
        if (n >= 4) {
            ans = (ans + f[n - 4]) % mod;
        }
        if (n >= 8) {
            ans = (ans + f[n - 8]) % mod;
        }
        return ans;
    }
};

Go

func numberOfWays(n int) int {
	const mod int = 1e9 + 7
	coins := []int{1, 2, 6}
	f := make([]int, n+1)
	f[0] = 1
	for _, x := range coins {
		for j := x; j <= n; j++ {
			f[j] = (f[j] + f[j-x]) % mod
		}
	}
	ans := f[n]
	if n >= 4 {
		ans = (ans + f[n-4]) % mod
	}
	if n >= 8 {
		ans = (ans + f[n-8]) % mod
	}
	return ans
}

TypeScript

function numberOfWays(n: number): number {
    const mod = 10 ** 9 + 7;
    const f: number[] = Array(n + 1).fill(0);
    f[0] = 1;
    for (const x of [1, 2, 6]) {
        for (let j = x; j <= n; ++j) {
            f[j] = (f[j] + f[j - x]) % mod;
        }
    }
    let ans = f[n];
    if (n >= 4) {
        ans = (ans + f[n - 4]) % mod;
    }
    if (n >= 8) {
        ans = (ans + f[n - 8]) % mod;
    }
    return ans;
}

Solution 2: Preprocessing + Dynamic Programming (Complete Knapsack)

We can start by preprocessing the number of ways to make up every amount from $1$ to $10^5$, and then return the corresponding number of ways based on the value of $n$:

  • If $n &lt; 4$, directly return $f[n]$;
  • If $4 \leq n &lt; 8$, return $f[n] + f[n - 4]$;
  • If $n \geq 8$, return $f[n] + f[n - 4] + f[n - 8]$.

Note the modulus operation for the answer.

The time complexity is $O(n)$, and the space complexity is $O(n)$. Where $n$ is the amount.

Python3

m = 10**5 + 1
mod = 10**9 + 7
coins = [1, 2, 6]
f = [0] * (m)
f[0] = 1
for x in coins:
    for j in range(x, m):
        f[j] = (f[j] + f[j - x]) % mod


class Solution:
    def numberOfWays(self, n: int) -> int:
        ans = f[n]
        if n >= 4:
            ans = (ans + f[n - 4]) % mod
        if n >= 8:
            ans = (ans + f[n - 8]) % mod
        return ans

Java

class Solution {
    private static final int MOD = 1000000007;
    private static final int M = 100001;
    private static final int[] COINS = {1, 2, 6};
    private static final int[] f = new int[M];

    static {
        f[0] = 1;
        for (int x : COINS) {
            for (int j = x; j < M; ++j) {
                f[j] = (f[j] + f[j - x]) % MOD;
            }
        }
    }

    public int numberOfWays(int n) {
        int ans = f[n];
        if (n >= 4) {
            ans = (ans + f[n - 4]) % MOD;
        }
        if (n >= 8) {
            ans = (ans + f[n - 8]) % MOD;
        }
        return ans;
    }
}

C++

const int m = 1e5 + 1;
const int mod = 1e9 + 7;
int f[m + 1];

auto init = [] {
    f[0] = 1;
    int coins[3] = {1, 2, 6};
    for (int x : coins) {
        for (int j = x; j < m; ++j) {
            f[j] = (f[j] + f[j - x]) % mod;
        }
    }
    return 0;
}();


class Solution {
public:
    int numberOfWays(int n) {
        int ans = f[n];
        if (n >= 4) {
            ans = (ans + f[n - 4]) % mod;
        }
        if (n >= 8) {
            ans = (ans + f[n - 8]) % mod;
        }
        return ans;
    }
};

Go

const (
    m   = 100001
    mod = 1000000007
)

var f [m]int

func init() {
    f[0] = 1
    coins := []int{1, 2, 6}
    for _, x := range coins {
        for j := x; j < m; j++ {
            f[j] = (f[j] + f[j-x]) % mod
        }
    }
}

func numberOfWays(n int) int {
    ans := f[n]
    if n >= 4 {
        ans = (ans + f[n-4]) % mod
    }
    if n >= 8 {
        ans = (ans + f[n-8]) % mod
    }
    return ans
}

TypeScript

const m: number = 10 ** 5 + 1;
const mod: number = 10 ** 9 + 7;
const f: number[] = Array(m).fill(0);

(() => {
    f[0] = 1;
    const coins: number[] = [1, 2, 6];
    for (const x of coins) {
        for (let j = x; j < m; ++j) {
            f[j] = (f[j] + f[j - x]) % mod;
        }
    }
})();

function numberOfWays(n: number): number {
    let ans = f[n];
    if (n >= 4) {
        ans = (ans + f[n - 4]) % mod;
    }
    if (n >= 8) {
        ans = (ans + f[n - 8]) % mod;
    }
    return ans;
}