Five students, including Amit, appear for an examination in which possible marks are integers between 0 and 50, both inclusive. The average marks for all the students is 38 and exactly three students got more than 32. If no two students got the same marks and Amit got the least marks among the five students, then the difference between the highest and lowest possible marks of Amit is
Explanation:
Total marks of all 5 students = 5 × 38 = 190
Case 1: Least marks of Amit. To minimize Amit’s marks we should maximize the marks of other 4 students. ⇒ 3 students get more than 32, hence their maximum marks can be 48, 49 and 50. [Marks are distinct]
2 students (including Amit) scored less than or equal to 32. ⇒ Maximum marks of 2nd student can be 32
∴ Sum of marks of these 4 students = 32 + 48 + 49 + 50 = 179 ⇒ Least marks of Amit = 190 – 179 = 11
Case 2: Maximum marks of Amit. Since Amit and one other student scored less than or equal to 32 marks, maximum marks Amit can score is 31. [Amit scored least and all scores are distinct.]
∴ Difference between highest and least score of Amit = 31 – 11 = 20
Hence, option (b).
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