This is a good question on the concepts of mean and minimization. Since the mean of the data set is fixed, the total is fixed as well. To minimize the number of students, we will have to reach this total, while keeping in mind that the smallest score any student obtained is 60.
Let the number of students be x. Since the mean score is 76, the sum of their scores is 76x.
Of the x students, 5 students scored 100, which totals to 500. Therefore, the total of the remaining students should be 76x – 500, with the smallest score being 60.
We can now plug in values for x from the answer options, starting with C.
If x = 12, the number of students who didn’t get a 100 = 7. Since each student got at least 60 points, the minimum total score by these 7 students is 420. The sum total of all the points = 500 + 420 = 920.
But 76 * 12 = 912. This is lesser than the minimum sum total required. This means, 12 is not the required minimum number of students.
If 12 is not the smallest possible number, 10 and 11 cannot be the smallest possible number. Options A, B and C can be eliminated.
If x = 13, total points = 76 * 13 = 988. Also, 500 + 8*60 = 500 + 480 = 980. This satisfies all the constraints defined in the question. So, the minimum possible number of students is 13.
Since, we are trying to minimize the number of students, while keeping the total points fixed, we also need to ensure that every student (other than the ones who scored 100) gets close to the minimum score, 60.
Hope this helps!
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