2585. Number of Ways to Earn Points

Description

There is a test that has n types of questions. You are given an integer target and a 0-indexed 2D integer array types where types[i] = [counti, marksi] indicates that there are counti questions of the ith type, and each one of them is worth marksi points.

    Return the number of ways you can earn exactly target points in the exam. Since the answer may be too large, return it modulo 109 + 7.

    Note that questions of the same type are indistinguishable.

    • For example, if there are 3 questions of the same type, then solving the 1st and 2nd questions is the same as solving the 1st and 3rd questions, or the 2nd and 3rd questions.

     

    Example 1:

    Input: target = 6, types = [[6,1],[3,2],[2,3]]
Output: 7
Explanation: You can earn 6 points in one of the seven ways:
- Solve 6 questions of the 0th type: 1 + 1 + 1 + 1 + 1 + 1 = 6
- Solve 4 questions of the 0th type and 1 question of the 1st type: 1 + 1 + 1 + 1 + 2 = 6
- Solve 2 questions of the 0th type and 2 questions of the 1st type: 1 + 1 + 2 + 2 = 6
- Solve 3 questions of the 0th type and 1 question of the 2nd type: 1 + 1 + 1 + 3 = 6
- Solve 1 question of the 0th type, 1 question of the 1st type and 1 question of the 2nd type: 1 + 2 + 3 = 6
- Solve 3 questions of the 1st type: 2 + 2 + 2 = 6
- Solve 2 questions of the 2nd type: 3 + 3 = 6

    Example 2:

    Input: target = 5, types = [[50,1],[50,2],[50,5]]
Output: 4
Explanation: You can earn 5 points in one of the four ways:
- Solve 5 questions of the 0th type: 1 + 1 + 1 + 1 + 1 = 5
- Solve 3 questions of the 0th type and 1 question of the 1st type: 1 + 1 + 1 + 2 = 5
- Solve 1 questions of the 0th type and 2 questions of the 1st type: 1 + 2 + 2 = 5
- Solve 1 question of the 2nd type: 5

    Example 3:

    Input: target = 18, types = [[6,1],[3,2],[2,3]]
Output: 1
Explanation: You can only earn 18 points by answering all questions.

     

    Constraints:

    • 1 <= target <= 1000
    • n == types.length
    • 1 <= n <= 50
    • types[i].length == 2
    • 1 <= counti, marksi <= 50

    Solutions

    Solution 1: Dynamic Programming

    We define $f[i][j]$ to represent the number of methods to get $j$ points exactly from the first $i$ types of questions. Initially, $f[0][0] = 1$, and the rest $f[i][j] = 0$. The answer is $f[n][target]$.

    We can enumerate the $i$th type of questions, suppose the number of questions of this type is $count$, and the score is $marks$. Then we can get the following state transition equation:

    $$ f[i][j] = \sum_{k=0}^{count} f[i-1][j-k \times marks] $$

    where $k$ represents the number of questions of the $i$th type.

    The final answer is $f[n][target]$. Note that the answer may be very large and needs to be modulo $10^9 + 7$.

    The time complexity is $O(n \times target \times count)$ and the space complexity is $O(n \times target)$. $n$ is the number of types of questions, and $target$ and $count$ are the target score and the number of questions of each type, respectively.

    Python Code
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    class Solution:
    def waysToReachTarget(self, target: int, types: List[List[int]]) -> int:
        n = len(types)
        mod = 10**9 + 7
        f = [[0] * (target + 1) for _ in range(n + 1)]
        f[0][0] = 1
        for i in range(1, n + 1):
            count, marks = types[i - 1]
            for j in range(target + 1):
                for k in range(count + 1):
                    if j >= k * marks:
                        f[i][j] = (f[i][j] + f[i - 1][j - k * marks]) % mod
        return f[n][target]

    Java Code
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    class Solution {
    public int waysToReachTarget(int target, int[][] types) {
        int n = types.length;
        final int mod = (int) 1e9 + 7;
        int[][] f = new int[n + 1][target + 1];
        f[0][0] = 1;
        for (int i = 1; i <= n; ++i) {
            int count = types[i - 1][0], marks = types[i - 1][1];
            for (int j = 0; j <= target; ++j) {
                for (int k = 0; k <= count; ++k) {
                    if (j >= k * marks) {
                        f[i][j] = (f[i][j] + f[i - 1][j - k * marks]) % mod;
                    }
                }
            }
        }
        return f[n][target];
    }
}

    C++ Code
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    class Solution {
public:
    int waysToReachTarget(int target, vector<vector<int>>& types) {
        int n = types.size();
        const int mod = 1e9 + 7;
        int f[n + 1][target + 1];
        memset(f, 0, sizeof(f));
        f[0][0] = 1;
        for (int i = 1; i <= n; ++i) {
            int count = types[i - 1][0], marks = types[i - 1][1];
            for (int j = 0; j <= target; ++j) {
                for (int k = 0; k <= count; ++k) {
                    if (j >= k * marks) {
                        f[i][j] = (f[i][j] + f[i - 1][j - k * marks]) % mod;
                    }
                }
            }
        }
        return f[n][target];
    }
};

    Go Code
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    func waysToReachTarget(target int, types [][]int) int {
	n := len(types)
	f := make([][]int, n+1)
	for i := range f {
		f[i] = make([]int, target+1)
	}
	f[0][0] = 1
	const mod = 1e9 + 7
	for i := 1; i <= n; i++ {
		count, marks := types[i-1][0], types[i-1][1]
		for j := 0; j <= target; j++ {
			for k := 0; k <= count; k++ {
				if j >= k*marks {
					f[i][j] = (f[i][j] + f[i-1][j-k*marks]) % mod
				}
			}
		}
	}
	return f[n][target]
}

    TypeScript Code
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    function waysToReachTarget(target: number, types: number[][]): number {
    const n = types.length;
    const mod = 10 ** 9 + 7;
    const f: number[][] = Array.from({ length: n + 1 }, () => Array(target + 1).fill(0));
    f[0][0] = 1;
    for (let i = 1; i <= n; ++i) {
        const [count, marks] = types[i - 1];
        for (let j = 0; j <= target; ++j) {
            for (let k = 0; k <= count; ++k) {
                if (j >= k * marks) {
                    f[i][j] = (f[i][j] + f[i - 1][j - k * marks]) % mod;
                }
            }
        }
    }
    return f[n][target];
}