Question

Suppose a set of students want to create a virtual study groups. The set consists of 86 students on the eastern timezone, and 47 students in non-eastern time zones. Of the students in the eastern timezone, only 12 are not in Boston.

i. Suppose each group is made of 2 students, one in the eastern time zone and one from noneastern timezone. How many possible study groups are there?

ii. Suppose each group is made of 3 students, one in Boston, one in the eastern time zone but not in Boston, and one from the non-eastern timezone. How many groups can we make?

iii.suggest the Boston students form study groups amongst themselves first. The groups must have at most 5 people for optimal studying. He then secures 3 rooms for these groups. Each room can be safely filled by at most 5 study groups. Show if the students can form the groups safely and the groups fit in these 3 rooms, then there must be at least 3 people in each group.

iv. With rooms set, the in-person study groups can begin. However, the Boston students want to let their remote fellows join via a virtual meetings. How many remote students are guaranteed to join the most welcoming in-person group?

show your steps, please

Answer 1

Students on the eastern timezone = S_E = 86

Students on the non-eastern timezone = S_NE = 47

From S_E Students not in Boston= S_NB = 12

From S_E Students in Boston= S_B = 86-12=74

i. If group is made of 2 students, one in the eastern time zone and one from noneastern timezone. Then there

Students on the eastern timezone = S_E = 86

Students on the non-eastern timezone = S_NE = 47

So the number of groups =47 (Because the non-eastern timezone Students = 47)

If 47 groups are formed, then no student from non-eastern timezone will remain,

ii. Suppose each group is made of 3 students, one in Boston, one in the eastern time zone but not in Boston, and one from the non-eastern timezone. How many groups can we make?

Students in Boston= S_B = 86-12=74

Student in the eastern time zone but not in Boston= S_NB = 12

Students on the non-eastern timezone = S_NE = 47

If we form Group of 3 Students; one in Boston, one in the eastern time zone but not in Boston, and one from the non-eastern timezone, then we can from 12 groups , because there are only 12 Student in the eastern time zone but not in Boston. Hence , if 12 groups are formed, the Student in the eastern time zone but not in Boston will finish.

iii.suggest the Boston students form study groups amongst themselves first. The groups must have at most 5 people for optimal studying. He then secures 3 rooms for these groups. Each room can be safely filled by at most 5 study groups. Show if the students can form the groups safely and the groups fit in these 3 rooms, then there must be at least 3 people in each group.

Students in Boston= S_B =74

Maximum number of students in a group=5

Number of rooms=3

Maximum number of groups that can be filled in a room=5

Total number of groups = 3*5=15

If we divide the 74 students in 15 groups then =74/15

=4 and 14 as reminder

If we fill the remaining 14 in first 14 groups, then the 14 groups will contain 5 students and one group of 4 students

Group	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Maximum /Group	5	5	5	5	5	5	5	5	5	5	5	5	5	5	4
Tot	5	10	15	20	25	30	35	40	45	50	55	60	65	70	74

If we reduce the number of Students in any group to 3 or less than three, then the removed students need to be added to other groups. But the first 14 groups contain maximum numbers, Hence addition is not possible. We can add one student in the last group, hence it will reach maximum number(5).then the rest of the removed students can not be added. If we think of creating a new group, the number of total groups will exceed the limit that can be safely filled in 3 rooms.

Hence it is proved.

iv. With rooms set, the in-person study groups can begin. However, the Boston students want to let their remote fellows join via a virtual meetings. How many remote students are guaranteed to join the most welcoming in-person group?

show your steps, please

From S_E Students in Boston= S_B = 74

Number of Rooms=3

Number of total groups=15

Maximum number of students in a group=5

Then the remaining Boston Students for a group =74-5=69

Number of groups that will let their remote fellows join via a virtual meetings=15-1=14

Number of remote students are guaranteed to join the most welcoming in-person group=69