A market-research firm surveyed 300 households in a town about three video-streaming services – Xstream, Yonder and Zapp. Each household reported which of the three services it subscribes to (possibly none). The following facts emerged:
- 105 households subscribe to Xstream, 100 to Yonder and 100 to Zapp.
- 10 households subscribe to all three services.
- 60 households subscribe to exactly two of the three services.
- The number of households subscribing to Yonder and Zapp but not Xstream is 5 more than the number subscribing to Xstream and Yonder but not Zapp.
- The number of households subscribing to Xstream and Zapp but not Yonder is 5 less than the number subscribing to Xstream and Yonder but not Zapp.
- 105 households subscribe to Xstream, 100 to Yonder and 100 to Zapp.
- 10 households subscribe to all three services.
- 60 households subscribe to exactly two of the three services.
- The number of households subscribing to Yonder and Zapp but not Xstream is 5 more than the number subscribing to Xstream and Yonder but not Zapp.
- The number of households subscribing to Xstream and Zapp but not Yonder is 5 less than the number subscribing to Xstream and Yonder but not Zapp.
Question 1
Numerical answer
How many of the surveyed households subscribe to none of the three services?
Show answer and solution
Answer: 75
Let the 'exactly two' regions be XY-only = d, YZ-only = e, XZ-only = f. Then e = d + 5, f = d − 5 and d + e + f = 60 → 3d = 60 → d = 20, e = 25, f = 15.
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Answer: 75.
Check: By inclusion–exclusion, union = ΣA − Σ(pairwise) + all three, where each pairwise intersection includes the triple: (105 + 100 + 100) − (30 + 35 + 25) + 10 = 225.
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Answer: 75.
Check: By inclusion–exclusion, union = ΣA − Σ(pairwise) + all three, where each pairwise intersection includes the triple: (105 + 100 + 100) − (30 + 35 + 25) + 10 = 225.
Question 2
Single correct option
How many households subscribe to exactly one of the three services?
- A. 145
- B. 150
- C. 155
- D. 165
Show answer and solution
Answer: (C) 155
Let the 'exactly two' regions be XY-only = d, YZ-only = e, XZ-only = f. Then e = d + 5, f = d − 5 and d + e + f = 60 → 3d = 60 → d = 20, e = 25, f = 15.
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Exactly one = 60 + 45 + 50 = 155.
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Exactly one = 60 + 45 + 50 = 155.
Question 3
Single correct option
How many households subscribe to Zapp but not to Yonder?
- A. 50
- B. 60
- C. 65
- D. 75
Show answer and solution
Answer: (C) 65
Let the 'exactly two' regions be XY-only = d, YZ-only = e, XZ-only = f. Then e = d + 5, f = d − 5 and d + e + f = 60 → 3d = 60 → d = 20, e = 25, f = 15.
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Zapp but not Yonder = only Z + (X & Z only) = 50 + 15 = 65. (The triple region is excluded because those households also have Yonder.)
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Zapp but not Yonder = only Z + (X & Z only) = 50 + 15 = 65. (The triple region is excluded because those households also have Yonder.)
Question 4
Single correct option
What is the ratio of households subscribing only to Xstream to households subscribing to Xstream along with at least one other service?
- A. 4 : 3
- B. 3 : 2
- C. 2 : 1
- D. 6 : 5
Show answer and solution
Answer: (A) 4 : 3
Let the 'exactly two' regions be XY-only = d, YZ-only = e, XZ-only = f. Then e = d + 5, f = d − 5 and d + e + f = 60 → 3d = 60 → d = 20, e = 25, f = 15.
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Only Xstream = 60. Xstream with at least one other = 20 + 15 + 10 = 45.
Ratio = 60 : 45 = 4 : 3.
Only X = 105 − 20 − 15 − 10 = 60; only Y = 100 − 20 − 25 − 10 = 45; only Z = 100 − 25 − 15 − 10 = 50.
| Region | Households |
|---|---|
| Only X | 60 |
| Only Y | 45 |
| Only Z | 50 |
| X & Y only | 20 |
| Y & Z only | 25 |
| X & Z only | 15 |
| All three | 10 |
| None | 75 |
At least one = 155 + 60 + 10 = 225, so none = 300 − 225 = 75.
Only Xstream = 60. Xstream with at least one other = 20 + 15 + 10 = 45.
Ratio = 60 : 45 = 4 : 3.
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