What is the minimum number of students enrolled in both G and L but not in K?

Let 'x' be the number of students enrolled in all three sports. Then the number of students enrolled only in L = 2x

It is given that there are a total of 17 students enrolled in G. Also, ten students enrolled in G are also enrolled in at least one more sport. Hence, the number of students enrolled in only G = 17 - 10 = 7

The number of students enrolled only in G is one less than the number of students enrolled only in L. Hence, the number of students enrolled only in L = 7+1 

$$\Rightarrow$$ 2x = 8 

$$\Rightarrow$$ x = 4

Let us assume that 'y' students are enrolled in K and L but not G. Then, the number of students enrolled only in K = y + 4

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 = 10 - 4 - z = 6 - z

It is given that a total of 39 students in the sports. 

7 + z + 4 + 6 - z + 8 + y + y + 4 = 39 

$$\Rightarrow$$ y = 5 

Number of students enrolled in G = 17 

Number of students enrolled in K = 9 + 4 + 5 + z = 18 + z

Number of students enrolled in L = 6 - z + 4 + 5 + 8 = 23 - z

It is given that the maximum student enrollment is in L.

$$\Rightarrow$$ 23 - z > 18 + z

$$\Rightarrow$$ 2z < 5

$$\Rightarrow$$ z < 2.5 

Therefore, we can say that z can take three values = {0, 1, 2}

The number of students  enrolled in both G and L but not in K = 6 - z. This number will be minimum when 'z' is maximum. z_{max} = 2

Therefore, the minimum number of students enrolled in both G and L but not in K = 6 - 2 = 4