Question: Later, the college imposed a condition that if after the change of electives, the enrollment in any elective (other than E7) dropped to less than 20 students, all the students who had left that course will be required to re-enroll for that elective.
Which of the following is a correct sequence of electives in decreasing order of their final enrollments?
Explanation:
Let us first try and tabulate the number of people for each of the electives before and after the change in elective
After Change in electives
Before Change
in Elective
E1 E2 E3 E4 E5 E6 Total
E1 9
5
10
1
4
2
31
E2
34
8
2
2
E3 2
6
25
2
E4
3
2
14
4
E5
5
30
E6
7
3
2
9
E7 4
16
30
5
5
41
101
Total
76
ow we know that after change of electives number of people choosing E2 increases by 30. This means that before change of electives, number of people choosing E2 is 76 – 30 = 46.
Now if we check row E2, the values 34, 8, 2 and 2 add upto 46. Since the values of blank cells can be 0 or 1, it means that the remaining 2 cells in row E2 will be 0.
Now before the change, E1 had 6 more students than E4, which means E4 had 31 – 6 = 25 students. If we look at the row E4, there are 2 blank cells. Also, the total of the remaining 4 cells adds upto 3 + 2 + 14 + 4 = 23. As total of all 6 cells has to be 25, it would mean that remaining 2 cells can only have a value of 1. Also the number of students before change of elective in E6 is 25 – 2 = 23. Again in row E6, sum of 4 occupied cells = 7 + 3 + 2 + 9 = 21. Here too with only 2 vacant cells, the balance 2 cells will have a value of 1 each. Now in E2 there are 10 students more than E3 before change of electives, which means in row E3 the total will be 36. This further implies that in row E5, total will be 300 – (31 + 46 + 36 + 25 + 23 + 101) = 38
After Change in electives
Before Change
in Elective
E1 E2 E3 E4 E5 E6 Total
E1 9
5
10
1
4
2
31
E2 0
34
8
0
2
2
46
E3 2
6
25
2
36
E4 1
3
2
14
4
25
E5
5
30
38
E6 1
7
3
1
2
9
23
E7 4
16
30
5
5
41
101
Total
76
So far, the total of columns E1 to E6 is 17 + 76 + 78 + 21 + 44 + 60 = 296
This means that the remaining 6 vacant cells in the table will be filled up four ‘1’s and two 0’s. Since the number of students after the reshuffle in E1 is 3 less than the number of students in E4 (which is 21 without the 2 blank cells), the total of column E1 has to be at least 18 which means the vacant cell in column E1 will be filled with 1. This further implies that the balance 2 cells in column E4 will be filled with 0. So then the remaining cells in column E3, E5 and E6 will be filled with 1. Our final table will appear as below.
After Change in electives
Before Change
in Elective
E1 E2 E3 E4 E5 E6 Total
E1 9
5
10
1
4
2
31
E2 0
34
8
0
2
2
46
E3 2
6
25
0
1
2
36
E4 1
3
2
14
1
4
25
E5 1
5
1
0
30
1
38
E6 1
7
3
1
2
9
23
E7 4
16
30
5
5
41
101
Total 18
76
79
21
45
61
300
Now using this table let us answer the questions
It is only for elective E1 that the enrollments have fallen to below 20. If the students who had left the course E1, now re-enroll for it, the final number of students for each of the electives will be as follows
After Change in electives
Before Change
in Elective
E1 E2 E3 E4 E5 E6 Total
E1 31
0
0
0
0
0
31
E2 0
34
8
0
2
2
46
E3 2
6
25
0
1
2
36
E4 1
3
2
14
1
4
25
E5 1
5
1
0
30
1
38
E6 1
7
3
1
2
9
23
E7 4
16
30
5
5
41
101
Total 40
71
79
20
41
59
300
As can be seen from the table the correct sequence of the electives in decreasing order of their electives is E2, E3, E6, E5, E1 and E4.
Hence, option (a).