Solution
Understanding the set :-
This is a Venn diagram set.
- Weβre told:
a) There are 39 students total, and each is in at least one sport.
b) Some are in only one sport, some in exactly two sports, and some in all three.
c) The clues give relationships between these groups.
- Draw the Venn diagram and,
Think of three overlapping circles labeled G, K, L.
That gives 7 possible regions:
a) Only G
b) Only K
c) Only L
d) G & K only
e) G & L only
f) K & L only
g) All three
- The approach.
a) Represent each region by a letter (say π, b, π, π, π, π, π).
b) Write equations for each clue.
c) Use totals to find the numbers.
Clue 1 says,
The number of students enrolled only in L is double the number of students enrolled in all the three sports.
Let the number of students enrolled in all the three sports be "x".
=> Number of students enrolled in only L will be "2x".
Clue 2 says,
There are a total of 17 students enrolled in G.
Also, clue 6 says,
Ten students enrolled in G are also enrolled in at least one more sport.
=> Therefore, the number of students enrolled in only G =
Clue 3 says,
The number of students enrolled only in G is one less than the number of students enrolled only in L.
We know, the number of students enrolled in only G = 7.
=> the number of students enrolled only in L =
=> 2x = 8
=> X = 4
Clue 4 says,
The number of students enrolled only in K is equal to the number of students who are enrolled in both K and L.
Here the number of students enrolled in K and L = Number of students enrolled in only K + L and number of students enrolled in all the three.
Now, Let the number of students enrolled in only K + L = y
=> the number of students enrolled in K and L = y + 4
=> number of students enrolled only in K = y + 4
Now, Let us assume that 'z' be the the number of students enrolled in G and K but not L.
Then, the number of students enrolled G and L bot not K =
=
Now, we know that there are total of 39 enrollments.
=>
β
Now we know,
Number of students enrolled in G = 17
Number of students enrolled in K =
Number of students enrolled in L =
clue 5 says,
The maximum student enrollment is in L.
β
β
β
Therefore, we can say that z can take three values = {0, 1, 2}
It is given that after withdrawal the number of students enrolled in K went down by one. This one student must have left sports K.
Hence we can say that the remaining 3 students must have left either G or L.
Before withdraw there were a total of 24 students were enrolled in exactly 1 sports, 11 students were enrolled in exactly 2 courses and 4 students were enrolled in all three courses.
The students which were enrolled in all three sports, withdrew from one of the sports.
Hence, we can say that now the number of students who were enrolled in exactly 2 courses = 11 + 4 = 15.
It is given that the number of students enrolled in G was six less than the number of students enrolled in L.
Let 'a' be the number of students who were enrolled in G and K but not L.
Then, the number of students who were enrolled in L and K but not G = a + 5
Consequently, we can say that the number of students enrolled in G and L but not K =
Number of students enrolled in this case =.
We can see that '14+2a' is an even number.
It is given that the number of students enrolled in K went down by one.
Therefore, we can say that the number of students enrolled in K earlier was an odd number.
Hence, the number of students enrolled in K = 18 + z = {18, 19, 20}
We can see that only '19' is an odd number.
Hence, we can say that the number of students enrolled in K after withdrawal = 18