To find out how many different ways four students can line up against a blackboard, we need to calculate the number of permutations of these four students.
When arranging [tex]\( n \)[/tex] students in a line, the number of possible arrangements (permutations) can be found by using the factorial function, denoted as [tex]\( n! \)[/tex]. The factorial of a number [tex]\( n \)[/tex] is the product of all positive integers less than or equal to [tex]\( n \)[/tex].
For our problem, we have 4 students, so we need to find [tex]\( 4! \)[/tex] (4 factorial):
[tex]\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \][/tex]
Thus, the number of different ways four students can line up against a blackboard is [tex]\( 24 \)[/tex].
Therefore, the correct answer is:
(D) 24
