Question

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

• A. 18
• B. 19
• C. 17
• D. 22
• A. 6
• B. 7
• C. 5
• D. 8

Answer 1

Correct Answer: 4

The data can be structured as follows:

• \( f + g + d = 10 \)   (Given)
• \( g + e = b \)   (Given)
• Also, \( g = 7 = 2g - 1 \Rightarrow 2g = 8 \Rightarrow g = 4 \)
• \( f = 4 \), and then \( f + d = 6 \Rightarrow d = 2 \)
• Let \( a = 7 \), \( c = 8 \)
• Total = 39 → \( b + e = 39 - (G + c) = 14 \)
• Given \( g + 2e = 14 \), and \( g = 4 \) ⇒ \( 4 + 2e = 14 \Rightarrow e = 5 \)
• Then \( b = g + e = 4 + 5 = 9 \)

Since number of students in L is maximum, consider the three possible cases:

Case-wise Distribution

1. Case (i): G = 17, K = 20, L = 21, \( d = 2 \), \( f = 4 \)
2. Case (ii): G = 17, K = 19, L = 22, \( d = 1 \), \( f = 5 \)
3. Case (iii): G = 17, K = 18, L = 23, \( d = 0 \), \( f = 6 \)

From the above, the number of students in set G and L but not K is represented by \( f \).

Final Answer: 4

Answer 2

The problem requires determining the number of students enrolled in L based on given conditions.
The ratio of students enrolled in K to L is 19:22, and we need the number enrolled in L. Let's denote:

• The number of students enrolled in K = 19x 
• The number of students enrolled in L = 22x

From the description, we're told the total enrollment is 39 students.
 

Sport	Students
Gilli-danda (G)	17
K (Based on ratio)	19x
L (Based on ratio)	22x

Given that L has the maximum enrollment and makes up part of 39 students, satisfying these conditions, the number of students enrolled in L (22) fits all constraints correctly. Thus, in the scenario described, the correct answer is 22. The process follows naturally from the stipulations and calculations outlined here.

Answer 3

Let us define the following variables:

• \( x \): students enrolled only in L
• \( y \): students enrolled only in G
• \( z \): students enrolled only in K
• \( a \): students enrolled in both G and K
• \( b \): students enrolled in both G and L
• \( c \): students enrolled in both K and L
• \( d \): students enrolled in all three sports

Given:

1. \( x = 2d \)
2. \( y = x - 1 \Rightarrow y = 2d - 1 \)
3. \( y + b + a + d = 17 \Rightarrow b + a + 3d = 18 \quad \text{(i)} \)
4. \( z = c - 1 \quad \text{(withdrawal adjustment)} \)
5. \( x + y + z + a + b + c + d = 39 \)
6. \( b + a + d = 10 \quad \text{(from students in G enrolled in ≥1 more sport)} \)

Substitute Known Values:

From (5):
\[ x + y + z + a + b + c + d = 39 \] \[ 2d + (2d - 1) + (c - 1) + a + b + c + d = 39 \] \[ 5d + a + b + 2c = 41 \quad \text{(ii)} \]

From (i):

\[ b + a = 18 - 3d \quad \text{(iii)} \]

From (vi):

\[ b + a = 10 - d \quad \text{(iv)} \]

Equating (iii) and (iv):
\[ 18 - 3d = 10 - d \Rightarrow 8 = 2d \Rightarrow d = 4 \]

Back Substitution:

• \( x = 2d = 8 \)
• \( y = 2d - 1 = 7 \)
• From (iv): \( b + a = 6 \)
• From (vi): \( b + a + d = 10 \Rightarrow 6 + 4 = 10 \quad \text{✓} \)

Try \( a = 2 \Rightarrow b = 4 \)

Final Values:

• \( a = 2 \): Students in both G and K
• All conditions satisfied

Answer: 2 students are enrolled in both G and K.

Answer 4

To find the number of students enrolled in both G and L after the withdrawal, let's define the variables:
Let x be the number of students enrolled in all three sports.
 

1. Students only in L = 2x
    
2. Students in G = 17
    
3. Students only in G = 2x - 1 (since it's one less than only in L)
    
4. Students only in K = Students in both K and L
    
5. Total G_LK (number of students in G and L only) = 17 - (10 + 2x - 1) = 8 - x
    
6. Before withdrawal:
   • It is given that maximum enrollment is in L. Calculate students in L:
   • Total G (enrollment in G) = 17
   • From student overlap, G+L = 8, G+K = 10 - x (since 10 students in G are in at least one more sport), G+L+K = x
7. After withdrawal, G = L - 6 and K = K - 1
8. Solving: Since G + L = 39 initially, L = (39 - 8) / 2 = 9.5, contradicts natural numbers, solving by equation, we get x = 3, thus 2x = 6

Number of students enrolled in both G and L: 6