Explanation:

Given:\nTotal number of students in the class ($N$) = $50$.\nMithilesh's rank from the top = $8^{\text{th}}$.\nHimadri's rank from the bottom = $20^{\text{th}}$. \nFirst, we need to determine if there is any overlap in their positions. The sum of their ranks from opposite ends is $8 + 20 = 28$.\nSince the sum of ranks ($28$) is less than the total number of students ($50$), there is no overlap between Mithilesh and Himadri; they are distinct individuals, and there are students between them. \nThe number of students between Mithilesh and Himadri can be calculated using the formula:\nNumber of students between = Total students - (Mithilesh's rank from top + Himadri's rank from bottom)\nNumber of students between = $50 - (8 + 20)$\nNumber of students between = $50 - 28$\nNumber of students between = $22$. \nAlternatively, consider the students accounted for by their positions:\nStudents from the top up to Mithilesh (including Mithilesh) = $8$.\nStudents from the bottom up to Himadri (including Himadri) = $20$.\nTotal students accounted for if there's no overlap = $8 + 20 = 28$.\nThe remaining students are those who are positioned between Mithilesh and Himadri.\nRemaining students = Total students - (Students accounted for)\nRemaining students = $50 - 28 = 22$. \nLet's evaluate the options:\nOption (1) $20$: Incorrect.\nOption (2) $21$: Incorrect.\nOption (3) $22$: This matches our calculated number of students.\nOption (4) $23$: Incorrect. \nTherefore, there are $22$ students between Mithilesh and Himadri. The correct option is (3) 22.