Question

Suppose that insertion sort is applied to the array \([1,2,3,5,7,9,11,x,15,13]\) and it takes exactly 2 swaps to sort the array. Select all possible values of \( x \).

• A. 14
• B. 16
• C. 12
• D. 10

Hint:

Insertion sort performs efficiently when elements are nearly sorted, as it only requires a small number of swaps to place misplaced elements in their correct positions.

Answer 1

The Correct Option is A

Insertion sort builds a sorted sequence incrementally from left to right, shifting elements as needed to place them in their correct positions. The number of swaps required depends on how far an element must move backward to reach its proper place.
To determine the number of swaps: 1. The given array is \([1,2,3,5,7,9,11,x,15,13]\). 2. The element \( x \) must be positioned correctly using insertion sort. 3. The final sorted order should be \([1,2,3,5,7,9,10,11,13,15]\) for some values of \( x \). 4. If \( x = 10 \), it must shift back two positions to reach its correct place. 5. Similarly, if \( x = 14 \), it also moves back two places, requiring exactly two swaps. Thus, the possible values of \( x \) that result in exactly two swaps are 10 and 14.
Conclusion: The values of \( x \) that require exactly two swaps in insertion sort are 10 and 14.