A dormitory has 40 students---12 sophomores, 8 juniors, and 20 seniors. Which of the following is equal to the number of ways to put all 40 in a row for a picture, with all 12 sophomores on the left, all 8 juniors in the middle, and all 20 seniors on the right?

Answer:The number of ways is equal to [tex]12!8!20![/tex]Step-by-step explanation:The multiplication principle states that If a first experiment can happen in n1 ways, then a second experiment can happen in n2 ways ... and finally a i-experiment can happen in ni ways therefore the total ways in which the whole experiment can occur aren1 x n2 x ... x ni Also, given n-elements in which we want to put them in a row, the total ways to do this are n! that is n-factorial.For example : We want to put 4 different objects in a row.The total ways to do this are [tex]4!=4.3.2.1=24[/tex] ways.Using the multiplication principle and the n-factorial number :The number of ways to put all 40 in a row for a picture, with all 12 sophomores on the left,all 8 juniors in the middle, and all 20 seniors on the right are : The total ways to put all 12 sophomores in a row multiply by the ways to put the 8 juniors in a row and finally multiply by the total ways to put all 20 senior in a row ⇒ [tex]12!8!20![/tex]