Number of Ways to Earn Points

To find the number of ways you can earn exactly the target points in the exam, we can utilize dynamic programming. We will use a 1D array dp, where dp[j] will represent the number of ways to achieve j points. The solution to this problem can be implemented as follows:

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class Solution:
    def waysToReachTarget(self, target: int, types: List[List[int]]) -> int:
        MOD = 10**9 + 7

        # Initialize dp array with zeros, and set the 0th index to 1 as there's one way to get 0 points
        dp = [0] * (target + 1)
        dp[0] = 1

        # Iterate through each type of question
        for count, marks in types:
            # Update dp array for each type of question
            for i in range(target, -1, -1):
                for j in range(1, count + 1):
                    next_points = i + marks * j
                    if next_points > target:
                        break
                    # Add the number of ways to achieve the points with the current question
                    dp[next_points] = (dp[next_points] + dp[i]) % MOD

        return dp[target]

The above code snippet defines a class Solution with a function waysToReachTarget, which takes a target integer and a 2D integer array types. The code then calculates the number of ways to reach the target points and returns the result modulo (10^9 + 7).

The complexity of the solution is (O(n \cdot t \cdot m)), where (n) is the number of types, (t) is the target, and (m) is the maximum count of questions in a type.

10 Prerequisite LeetCode Problems

This is a combination of dynamic programming and combinatorics. As it involves reaching a certain target using a certain set of values, this is a type of knapsack problem where items can be used more than once (unbounded knapsack). Here are some problems for this particular problem:

Coin Change: This problem requires you to compute the fewest number of coins that you need to make up a certain amount of money.
Combination Sum: This problem requires you to find all possible combinations of candidates that sum up to a target.
Combination Sum II: This is a similar problem but the candidates can only be used once.
Combination Sum III: This is also similar but you need to find all possible combinations of candidates that sum up to a target using exactly k numbers.
Combination Sum IV: This problem requires you to find the number of possible combinations that sum up to a target.
Perfect Squares: This problem is similar to the Coin Change problem but with a twist. It requires you to compute the least number of perfect square numbers that sum to a certain number.
Partition Equal Subset Sum: This problem is a variant of the subset sum problem and requires you to determine if a set can be partitioned into two subsets such that the sum of elements in both subsets is the same.
0/1 Knapsack: This problem is a classic example of a knapsack problem, where you need to find the maximum value that can be achieved with a certain weight limit.
Coin Change 2: This problem requires you to find the number of ways to make up a certain amount of money with a set of coins.
Unbounded Knapsack: Although the exact problem is not there on LeetCode, understanding the concept of Unbounded Knapsack is crucial for such problems. In these problems, you can use one item multiple times.

These cover dynamic programming and how to apply it in different situations. They will also familiarize you with the knapsack problem and its variants, which seem to be closely related to your original problem.

Problem Classification

Problem Statement: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

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.