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 the1st
and2nd
questions is the same as solving the1st
and3rd
questions, or the2nd
and3rd
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.
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