Swift Coding Test Course, Understanding Combinations

In this course, we will explore the concept of combinations and how to utilize it in preparation for Swift coding tests for employment. A combination refers to the way of selecting r elements from a given set of n elements. This can help solve various algorithmic problems.

Definition of Combinations

Combinations are used when the order of selection does not matter. For example, when selecting 2 elements from {A, B, C}, {A, B} and {B, A} are considered the same combination. Mathematically, combinations can be expressed as follows:

C(n, r) = n! / (r! * (n - r)!)

Here, ! denotes factorial, and n! represents the product of all positive integers up to n.

Problem Description

Now, let’s look at a problem that utilizes combinations:

Problem: Sum of Combinations

For given integers n and k, write a function countCombinations(n: Int, k: Int, target: Int) -> Int to find the number of combinations of selecting k elements from the numbers from 1 to n such that their sum equals target.

Input

• n: Integer (1 ≤ n ≤ 20)
• k: Integer (1 ≤ k ≤ n)
• target: Integer (1 ≤ target ≤ 100)

Output

Print the number of combinations.

Problem-Solving Process

Now let’s explore the step-by-step process to analyze and solve the problem.

Step 1: Problem Analysis

This problem involves finding combinations of selecting k elements from the given n numbers such that their sum equals target. Since the selection does not depend on the order, combinations should be used. Therefore, it is essential to first understand and implement the concept of combinations.

Step 2: Combination Generation Algorithm

To generate combinations, a backtracking technique using recursion is commonly employed. This allows us to determine the current selected numbers and the range of numbers to be used next. In particular, each time a number is selected, it is necessary to check if that number has been chosen.

Step 3: Code Implementation

Below is an example code for a combination generator that includes the method to check if the sum of selected specific numbers equals target:

func countCombinations(n: Int, k: Int, target: Int) -> Int {
    var count = 0
    var combination = [Int]()
    
    func backtrack(start: Int, k: Int, sum: Int) {
        if k == 0 {
            if sum == target {
                count += 1
            }
            return
        }
        
        for i in start...n {
            combination.append(i)
            backtrack(start: i + 1, k: k - 1, sum: sum + i)
            combination.removeLast()
        }
    }
    
    backtrack(start: 1, k: k, sum: 0)
    return count
}

Step 4: Code Explanation

Let’s take a look at the above code:

• countCombinations: This method counts the total number of combinations. It starts the combination generation by passing initial values to the backtrack method.
• backtrack: Called recursively, it adds a specific number to the current combination and selects the next number.
• Conditional Statement: If k is 0, it checks if the combination of the currently selected numbers matches target.

Step 5: Testing

Now let’s validate the function with several test cases:

print(countCombinations(n: 5, k: 2, target: 5))  // Output: 1 (Combination: [2, 3])
print(countCombinations(n: 7, k: 3, target: 10)) // Output: 4 (Combinations: [1, 2, 7], [1, 3, 6], [2, 3, 5], [1, 4, 5])
print(countCombinations(n: 10, k: 2, target: 8)) // Output: 3 (Combinations: [2, 6], [3, 5], [1, 7])

Conclusion

In this lecture, we explored the process of implementing combinations in Swift and solving the problem of finding the number of combinations that satisfy specific conditions. Understanding algorithms like combinations is crucial for coding tests in job interviews, so consistent practice is recommended.

Continue to tackle various algorithmic problems to enhance your skills. Thank you!