A hostel has 100 residents. The hostel runs three clubs – Drama, Music and Sports – and a resident may join any number of them. The following facts are known about club membership:
1. Exactly 30 residents are not members of any club.
2. The Drama, Music and Sports clubs have 27, 33 and 37 members respectively.
3. Exactly 19 residents are members of exactly two clubs.
4. The number of residents who are members of Sports only is 5 more than the number who are members of Music only.
5. The number of residents who are members of Drama only is twice the number who are members of Drama and Music but not Sports.
1. Exactly 30 residents are not members of any club.
2. The Drama, Music and Sports clubs have 27, 33 and 37 members respectively.
3. Exactly 19 residents are members of exactly two clubs.
4. The number of residents who are members of Sports only is 5 more than the number who are members of Music only.
5. The number of residents who are members of Drama only is twice the number who are members of Drama and Music but not Sports.
Question 1
Numerical answer
How many residents are members of all three clubs?
Show answer and solution
Answer: 4
Let g = members of all three. Counting memberships: 27 + 33 + 37 = 97 = (exactly one) + 2 × (exactly two) + 3g, and (exactly one) + (exactly two) + g = 70.
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
Question 2
Single correct option
How many residents are members of exactly one club?
- A. 43
- B. 49
- C. 47
- D. 51
Show answer and solution
Answer: (C) 47
Let g = members of all three. Counting memberships: 27 + 33 + 37 = 97 = (exactly one) + 2 × (exactly two) + 3g, and (exactly one) + (exactly two) + g = 70.
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
12 + 15 + 20 = 47.
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
12 + 15 + 20 = 47.
Question 3
Single correct option
How many residents are members of the Music club but not of the Sports club?
- A. 15
- B. 25
- C. 21
- D. 29
Show answer and solution
Answer: (C) 21
Let g = members of all three. Counting memberships: 27 + 33 + 37 = 97 = (exactly one) + 2 × (exactly two) + 3g, and (exactly one) + (exactly two) + g = 70.
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
Music but not Sports = Music only + (Drama & Music only) = 15 + 6 = 21.
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
Music but not Sports = Music only + (Drama & Music only) = 15 + 6 = 21.
Question 4
Single correct option
How many residents are members of Drama or Music (or both) but not of Sports?
- A. 33
- B. 39
- C. 21
- D. 27
Show answer and solution
Answer: (A) 33
Let g = members of all three. Counting memberships: 27 + 33 + 37 = 97 = (exactly one) + 2 × (exactly two) + 3g, and (exactly one) + (exactly two) + g = 70.
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
Drama only + Music only + Drama & Music only = 12 + 15 + 6 = 33.
Check: (Drama ∪ Music) = 27 + 33 − (6 + 4) = 50; of these, those also in Sports = 5 + 8 + 4 = 17 → 50 − 17 = 33 ✓.
Subtracting: (exactly two) + 2g = 27 → 19 + 2g = 27 → g = 4, and exactly one = 70 − 19 − 4 = 47.
Let DM-only = d. Then Drama-only = 2d. Music-only = b, Sports-only = b + 5.
Singles: 2d + b + b + 5 = 47 → d + b = 21.
Music: b + d + (MS-only) + 4 = 33 → MS-only = 8.
DS-only = 19 − d − 8 = 11 − d. Drama: 2d + d + (11 − d) + 4 = 27 → d = 6.
| Region | Count |
|---|---|
| Drama only | 12 |
| Music only | 15 |
| Sports only | 20 |
| Drama & Music only | 6 |
| Music & Sports only | 8 |
| Drama & Sports only | 5 |
| All three | 4 |
| None | 30 |
Drama only + Music only + Drama & Music only = 12 + 15 + 6 = 33.
Check: (Drama ∪ Music) = 27 + 33 − (6 + 4) = 50; of these, those also in Sports = 5 + 8 + 4 = 17 → 50 − 17 = 33 ✓.
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