Problem Description

In the given LeetCode problem, there is an exam composed of n types of questions. The objective is to find the number of different ways to achieve a specific score, referred to as target. The questions are presented as a 2D array, types, where each element types[i] contains two values: count_i and marks_i. count_i tells us the number of available questions of the ith type, while marks_i tells us the points awarded for each question of that type.

The challenge lies in combining these questions in such a way that the sum of the points equals the target score. A key detail to consider is that all questions of the same type are identical in terms of points and cannot be distinguished by sequence. The result should be calculated modulo ( 10^9 + 7 ) to handle large numbers.

For example, if you have three types of questions, each type offering 1, 2, and 3 points and the target is 5, you must figure out all possible combinations of those questions that add up to exactly 5 points.

Intuition

The solution uses a dynamic programming (DP) approach to tackle this problem methodically. DP is a strategy often used to solve counting problems like this, particularly when the problem involves making choices at several steps, and the objective is to count the total number of ways to achieve a certain outcome.

We define a DP matrix f where each entry f[i][j] represents the number of ways to score j points using only the first i types of questions. The dynamic programming table is filled in a bottom-up manner using the recurrence relation described in the solution approach.

To calculate the number of ways f[i][j], we iterate through each possible number of questions we could answer of the ith type (from 0 to count[i]) and add the ways from the previous iteration (i-1) that would lead to the current score (j). This accumulation process ultimately gives us the number of ways to reach every potential score up to the target, using the available questions.

By carefully iterating over the candidates and updating our dynamic programming table, we can progressively build up the solution to include all types of questions. In the end, f[n][target] will give us the exact number of ways one could achieve the target score, where n is the total number of question types available in the exam.

Learn more about Dynamic Programming patterns.

Solution Approach

To implement the solution, we use a two-dimensional list (or array in some languages) f of size (n + 1) * (target + 1) as our dynamic programming table, where n is the number of question types.

This DP table f allows us to store intermediate results: f[i][j] will hold the number of ways to score exactly j points with the first i types of questions. The process begins by initializing f[0][0] to 1, which represents the fact that there is one way to score 0 points without any questions. All other entries in f start at 0, as initially, there are no known ways to reach scores higher than 0.

The solution iterates over each type of question i using a nested for loop. The outer loop runs through the question types, and the inner loops go through the target scores (from 0 to target) and the number of questions of the current type (from 0 to count[i]). For each question type, it updates the DP table entries for all possible scores j up to target.

Here is the key part of the algorithm:

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

In the above code block:

• count and marks are extracted from the types array for the ith question type.
• We need to check if the current target score j can be reached by answering k questions of type i, which is possible if j is greater than or equal to k * marks.
• If this condition is satisfied, we retrieve the count of ways to achieve the remaining score (j - k * marks), which is found in f[i-1][j - k * marks]. We add this count to f[i][j] to include the additional ways facilitated by the current question type.
• As the number of combinations can be very large, we use the modulo operation with mod = 10**9 + 7 to avoid integer overflow and to keep the numbers within the specified limit.

After filling out the DP table, the final answer, which is the number of ways to reach exactly target points, is found at f[n][target].

This dynamic programming approach provides an efficient way to solve the counting problem by breaking it down into smaller sub-problems and building up the solution incrementally.

Example Walkthrough

Let's consider an example where we have n = 2 types of questions and the target score is 5. The available questions are given as types = [[2, 3], [1, 5]], meaning we have two type 1 questions with 3 points each, and one type 2 question with 5 points.

Our dynamic programming table f will be initialized to zero with dimensions (n + 1) * (target + 1), resulting in a 3x6 matrix since n = 2 and target = 5. The first row f[0] and the first column f[x][0] for all x are initialized to signify the number of ways to achieve a score of 0 with different available types. In other words, f[0][0] will be set to 1 because there is only one way to achieve a score of 0 (by not selecting any questions), and all other cells in the first column are set to 1 as they signify the number of ways to reach a score of 0 with x available types.

Our table looks like this after initialization:

f[ ][0][1][2][3][4][5]
f[0][1][0][0][0][0][0]
f[1][ ][ ][ ][ ][ ][ ]
f[2][ ][ ][ ][ ][ ][ ]

Now, we fill the table using the approach described earlier:

1. For i = 1, we have count = 2 and marks = 3. We iterate over j from 0 to 5 and k from 0 to 2. This gives us:

   • For j = 0: k can be 0, so f[1][0] = 1.
   • For j = 1, 2, 4: k can only be 0 since j is less than marks * k for k > 0.
   • For j = 3: k can be 0 or 1 (since 3 points can be achieved by one question of the first type), so f[1][3] = f[0][3] + f[0][0] (since f[0][3] is 0, f[1][3] becomes 1).
   • For j = 5: k can be 0 (impossible) or 1 (but we only have 3 points questions, so it's not possible either).

   The table now looks like this:

   f[ ][0][1][2][3][4][5]
   f[0][1][0][0][0][0][0]
   f[1][1][0][0][1][0][0]
   f[2][ ][ ][ ][ ][ ][ ]

2. For i = 2, count = 1 and marks = 5. We follow the same process for j from 0 to 5 and k from 0 to 1.

   • For j = 5: k can be 0 or 1 (since we can achieve 5 points with one question of the second type), so f[2][5] = f[1][5] + f[1][0] (since f[1][5] is 0, f[2][5] becomes 1).

   After filling the table, we have:

   f[ ][0][1][2][3][4][5]
   f[0][1][0][0][0][0][0]
   f[1][1][0][0][1][0][0]
   f[2][1][0][0][1][0][1]

In this example, the number of ways to achieve exactly target = 5 points is found in f[2][5], which is 1. Therefore, there is one way to achieve the target score using the available questions.

Solution Implementation

Time and Space Complexity

The algorithm consists of three nested loops: the outermost loop runs through each type of question (n types in total), the middle loop iterates through all possible target scores up to target, and the innermost loop iterates up to count + 1 times, where count represents the maximum number of questions of a given type.

The time complexity of this nested loop structure is O(n * target * count). This is because for each of the n types, we perform target + 1 iterations in the middle loop, and up to count + 1 iterations in the inner loop for each iteration of the middle loop.

The space complexity arises from the 2D array f, which has dimensions (n + 1) * (target + 1). Hence, the space complexity is O(n * target) since we need to store a computed value for every combination of question types and scores up to the target. The additional constants (+1) do not affect the asymptotic complexities and are therefore omitted in Big O notation.

Learn more about how to find time and space complexity quickly using problem constraints.