Let $P$ be a quicksort program to sort numbers in ascending order using the first element as the pivot. Let $t_1$ and $t_2$ be the number of comparisons made by $P$ for the inputs [1 2 3 4 5] and [4 1 5 3 2] respectively. Which one of the following holds?

When Quicksort uses the first element as the pivot:

1. Input 1:[1 2 3 4 5] is already sorted.

   • Partitioning an already sorted array of size $n$ using the first element results in the worst-case behavior with unbalanced partitions of sizes $0$ and $n-1$.

   • The number of comparisons made is:

     $$t_1 = (n-1) + (n-2) + \dots + 1 = 4 + 3 + 2 + 1 = 10$$

2. Input 2:[4 1 5 3 2]

   • First pivot is $4$, which partitions into elements smaller than $4$ ([1, 3, 2]) and elements greater than $4$ ([5]), requiring $4$ comparisons.

   • Sub-array [1, 3, 2] with pivot $1$ splits into [] and [3, 2], requiring $2$ comparisons.

   • Sub-array [3, 2] with pivot $3$ requires $1$ comparison.

   • Sub-array [5] has size 1 (0 comparisons).

   • Total comparisons $t_2 = 4 + 2 + 1 = 7$.

Since $t_1 = 10$ and $t_2 = 7$, we have $t_1 > t_2$. Hence, (C) is the correct option.