Question

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

• A. 10
• B. 12
• C. 14
• D. 16

Hint:

In insertion sort, the number of swaps corresponds to the number of elements that are out of order and need to be moved into the correct position in the sorted part of the array.

Answer 1

The Correct Option is A, C

The given array is \( [1, 3, 5, 7, 9, 11, x, 15, 13] \), and we are told that insertion sort takes exactly two swaps to sort the array. We need to find the values of \( x \) such that exactly two swaps are required.

1. Step-by-step analysis of insertion sort:
Insertion sort works by repeatedly selecting an element and inserting it into the correct position in the sorted part of the array. A swap occurs when the selected element is out of order compared to the element it should be placed next to.

2. Identifying the possible values of \( x \):
For \( x = 10 \), the array becomes \( [1, 3, 5, 7, 9, 10, 11, 15, 13] \). The insertion sort will need to move \( 10 \) after \( 9 \), and then after \( 11 \) (one swap between 10 and 11, and another swap to place 13 in the correct position).
For \( x = 12 \), the array becomes \( [1, 3, 5, 7, 9, 11, 12, 15, 13] \). The insertion sort will need to move \( 12 \) after \( 11 \) (one swap between 12 and 11, and another swap to place 13 correctly).
For \( x = 14 \), the array becomes \( [1, 3, 5, 7, 9, 11, 14, 15, 13] \). The insertion sort will need to move \( 14 \) after \( 11 \) (one swap between 14 and 11, and another swap to place 13 correctly).

In all of these cases, exactly two swaps are required, so the possible values of \( x \) are 10, 12, and 14.

Thus, the correct answer is (A) 10, (B) 12, (C) 14.