Count Ways to Distribute Candies

This problem can be solved using dynamic programming. We’ll create a DP table where dp[i][j] will represent the number of ways to distribute i candies into j bags, making sure that each bag has at least one candy.

The base case is that there’s only one way to distribute one candy into one bag. Then we’ll build up the solution by considering each candy and each bag.

We’ll use the formula:

dp[i][j] = j * dp[i - 1][j] + dp[i - 1][j - 1]

Here’s the code implementing the solution:

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class Solution:
    def waysToDistribute(self, n: int, k: int) -> int:
        MOD = 10**9 + 7
        
        # Initialize the DP table
        dp = [[0] * (k + 1) for _ in range(n + 1)]
        dp[1][1] = 1

        # Iterate through the candies and bags
        for i in range(2, n + 1):
            for j in range(1, k + 1):
                dp[i][j] = (j * dp[i - 1][j] + dp[i - 1][j - 1]) % MOD

        return dp[n][k]

Explanation:

j * dp[i - 1][j] represents distributing the i-th candy into one of the existing j bags.
dp[i - 1][j - 1] represents distributing the i-th candy into a new bag, leaving the previous distribution unchanged.

This code calculates the number of different ways to distribute n candies into k bags, and returns the result modulo (10^9 + 7), as required by the problem statement.

Identifying Problem Isomorphism

“Count Ways to Distribute Candies” can be approximately mapped to “Combination Sum IV”.

Reasoning:

Both problems are about counting the number of ways to distribute items (candies or numbers) with specific constraints. In “Count Ways to Distribute Candies”, the goal is to find the number of ways to distribute candies to a group of people. In “Combination Sum IV”, the goal is to find the number of possible combinations that add up to a target number.

The key concept shared here is dynamic programming used to calculate the number of possible distributions (or combinations) under certain constraints.

“Combination Sum IV” is simpler as it deals with the direct sum of integers to reach a target, while “Count Ways to Distribute Candies” requires you to consider the additional constraint of distributing different candies among different people.

This involves dynamic programming, combinatorics and modulus arithmetic. Here are 10 problems to prepare:

“Unique Paths” (LeetCode Problem #62): This problem introduces you to the basic concepts of dynamic programming.
“Climbing Stairs” (LeetCode Problem #70): This problem is a basic and popular example of dynamic programming.
“Combination Sum IV” (LeetCode Problem #377): This is another example of a dynamic programming problem.
“Unique Paths II” (LeetCode Problem #63): This problem introduces the concept of dynamic programming with obstacles.
“Perfect Squares” (LeetCode Problem #279): This problem introduces dynamic programming with square numbers.
“Coin Change” (LeetCode Problem #322): This problem deals with minimum number of coins to make a certain amount, which requires understanding of dynamic programming.
“House Robber” (LeetCode Problem #198): This problem will give you an understanding of dynamic programming in sequence problems.
“Partition Equal Subset Sum” (LeetCode Problem #416): This problem is about partitioning a set into two subsets with equal sum, using dynamic programming.
“Pascal’s Triangle II” (LeetCode Problem #119): This problem introduces the combinatorics concept which is important for the main problem.
“Subarray Sum Equals K” (LeetCode Problem #560): This problem involves prefix sums and modulus arithmetic, which are used in the main problem.

You will have a good understanding of the techniques needed to solve the main problem, including dynamic programming, combinatorics, and modulus arithmetic.

The problem “1692. Count Ways to Distribute Candies” involves combinatorial calculations and dynamic programming. Here are 6 simpler problems that you might want to solve first in order to be better prepared for it:

“518. Coin Change 2” - It’s an extension of “Coin Change”, now dealing with combinations.
“343. Integer Break” - This problem is about distributing a number into several parts, which is similar to distributing candies.
“139. Word Break” - This problem gives a good practice for using DP with a twist of checking conditions.
“935. Knight Dialer” - This problem also involves counting ways with dynamic programming but with a different context.
“688. Knight Probability in Chessboard” - A DP problem that includes calculation of probabilities.
“1223. Dice Roll Simulation” - A problem involving counting sequences under constraints using DP.

By solving these problems, you should gain a good understanding of dynamic programming and combinatorics, which are both important for tackling problem “1692. Count Ways to Distribute Candies”.

Problem Classification

Problem Statement: There are n unique candies (labeled 1 through n) and k bags. You are asked to distribute all the candies into the bags such that every bag has at least one candy.

There can be multiple ways to distribute the candies. Two ways are considered different if the candies in one bag in the first way are not all in the same bag in the second way. The order of the bags and the order of the candies within each bag do not matter.

For example, (1), (2,3) and (2), (1,3) are considered different because candies 2 and 3 in the bag (2,3) in the first way are not in the same bag in the second way (they are split between the bags (2) and (1,3)). However, (1), (2,3) and (3,2), (1) are considered the same because the candies in each bag are all in the same bags in both ways.

Given two integers, n and k, return the number of different ways to distribute the candies. As the answer may be too large, return it modulo 109 + 7.

Example 1:

Input: n = 3, k = 2 Output: 3 Explanation: You can distribute 3 candies into 2 bags in 3 ways: (1), (2,3) (1,2), (3) (1,3), (2) Example 2:

Input: n = 4, k = 2 Output: 7 Explanation: You can distribute 4 candies into 2 bags in 7 ways: (1), (2,3,4) (1,2), (3,4) (1,3), (2,4) (1,4), (2,3) (1,2,3), (4) (1,2,4), (3) (1,3,4), (2)

Example 3:

Input: n = 20, k = 5 Output: 206085257 Explanation: You can distribute 20 candies into 5 bags in 1881780996 ways. 1881780996 modulo 109 + 7 = 206085257.

Constraints:

1 <= k <= n <= 1000

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?

Alternatively, if you’re working on a specific problem, you might ask something like:

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.