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Exercise 1

Write a recursive definition for generating all combinations of length k from an array of distinct numbers. Include a terminal condition and dead-end condition.

For the dead-end condition, think about what numbers we should or shouldn't add to our current combination to reach a length of k.

For example, imagine we have a partial combination of [2] when considering our example array of [1, 2, 3, 4] and k set to 2. We shouldn't create a combination of [2, 2]. This isn't a valid combination. Nor should we create a combination of [2, 1], as we've likely already found the combination [1, 2]. What numbers should we consider adding to our partial combination of [2]?

Solution

Terminal Condition: The problem description hints that we need to find all combinations of "size k." This means that we've reached the terminal condition whenever the candidate array has a length of k.

Dead-End Condition:

The dead-end condition is trickier in this problem. While it feels similar to the previous one, we can't simply apply the logic from the "Permutations" problem. If we just checked if the element was included, we'd get both [1,2] and [2,1], which is invalid. The same issue would occur if we tried to filter the array.

So, what's left?

The key insight is realizing that when we're looking at a given number, the subsequent numbers we can add to the combination are only those that come after it in the array. Not before, and not the same number.

Suppose we have an array of five elements [1, 2, 3, 4, 5] and we're looking for all combinations of length 3.

After we add 1 as a candidate, only numbers 2, 3, 4, and 5 are valid options. Similarly, after we add the number 2, we can no longer choose 1 and 2, but only 3, 4, and 5.

In other words, we can only move forward once we choose an element. We can achieve this implicitly by slicing the array from idx + 1 of the element, so that element and all elements before it are no longer an option.

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