Explanation:

The Quicksort algorithm is a divide-and-conquer sorting algorithm that works by selecting a pivot element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively.

In this question, we are given that there is a procedure for finding a pivot element that splits the list into two sub-lists, each of which contains at least one-fifth of the elements.

Key Observation:
The key observation here is that after selecting a pivot element, the partitioning step divides the array into two sub-arrays, one of which contains at least one-fifth of the elements. This means that the size of the sub-array containing at least one-fifth of the elements is at most 4/5 times the size of the original array.

Explanation of the Options:
Let's go through each option and see which one is correct:

a) T(n) = 2T(n/5) + n
This option suggests that the number of comparisons required to sort n elements is twice the number of comparisons required to sort n/5 elements, plus n. This is not correct because we are given that the pivot splits the list into two sub-lists, each containing at least one-fifth of the elements. So, the size of the sub-array containing at least one-fifth of the elements is at most 4/5 times the size of the original array. Therefore, the recurrence relation should have a factor of 4/5 instead of 2.

b) T(n) = T(n/5) + T(4n/5) + n
This option suggests that the number of comparisons required to sort n elements is the number of comparisons required to sort n/5 elements, plus the number of comparisons required to sort 4n/5 elements, plus n. This is the correct recurrence relation because after selecting a pivot element, we divide the array into two sub-arrays, one of which contains at least one-fifth of the elements.

c) T(n) = 2T(4n/5) + n
This option suggests that the number of comparisons required to sort n elements is twice the number of comparisons required to sort 4n/5 elements, plus n. This is not correct because we are given that the pivot splits the list into two sub-lists, each containing at least one-fifth of the elements. So, the size of the sub-array containing at least one-fifth of the elements is at most 4/5 times the size of the original array. Therefore, the recurrence relation should have a factor of 4/5 instead of 2.

d) T(n) = 2T(n/2) + n
This option suggests that the number of comparisons required to sort n elements is twice the number of comparisons required to sort n/2 elements, plus n. This is not correct because we are given that the pivot splits the list into two sub-lists, each containing at least one-fifth of the elements. So, the size of the sub-array containing at least one-fifth of the elements is at most 4/5 times the size of the original array. Therefore, the recurrence relation should have a factor of 4/5 instead of 2.

Conclusion: