D. An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130
use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use
Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. Find,
a. how many uses only Radio?
b. how many uses only Television?
c. how many use radio and television but not magazine?
d. how many use Television and Magazine but not radio?


We are given the following information:

- Total clients: \( 170 \)
- Clients using Television (T): \( 115 \)
- Clients using Radio (R): \( 110 \)
- Clients using Magazines (M): \( 130 \)
- Clients using both Television and Magazines (T ∩ M): \( 85 \)
- Clients using both Television and Radio (T ∩ R): \( 75 \)
- Clients using both Radio and Magazines (R ∩ M): \( 95 \)
- Clients using all three (T ∩ R ∩ M): \( 70 \)

To represent this data using a Venn diagram, we will define the overlapping regions, starting with those who use all three services, and work our way out to the regions where clients use only one or two services. 

Let's break it down step by step:

### 1. Draw the Venn Diagram
To construct the Venn diagram:
- Draw three intersecting circles, representing Television (T), Radio (R), and Magazines (M).
- The innermost region, where all three circles overlap, will represent those using all three services, i.e., \( T ∩ R ∩ M = 70 \).

Next, we use the inclusion-exclusion principle to calculate the regions.

### Step-by-Step Breakdown of Regions

#### Clients using all three services:
\[
T ∩ R ∩ M = 70
\]

#### Clients using two services but not all three:
- Television and Radio, but not Magazines:
\[
(T ∩ R) \setminus (T ∩ R ∩ M) = 75 - 70 = 5
\]
- Radio and Magazines, but not Television:
\[
(R ∩ M) \setminus (T ∩ R ∩ M) = 95 - 70 = 25
\]
- Television and Magazines, but not Radio:
\[
(T ∩ M) \setminus (T ∩ R ∩ M) = 85 - 70 = 15
\]

#### Clients using only one service:
Now, let's calculate the number of clients using only one service:

- Clients using only Television:
\[
T \setminus (T ∩ R \cup T ∩ M) = 115 - (15 + 5 + 70) = 115 - 90 = 25
\]
- Clients using only Radio:
\[
R \setminus (T ∩ R \cup R ∩ M) = 110 - (5 + 25 + 70) = 110 - 100 = 10
\]
- Clients using only Magazines:
\[
M \setminus (T ∩ M \cup R ∩ M) = 130 - (15 + 25 + 70) = 130 - 110 = 20
\]

### a. How many use only Radio?
From the above, the number of clients using only Radio is:
\[
10
\]

### b. How many use only Television?
From the above, the number of clients using only Television is:
\[
25
\]

### c. How many use Radio and Television but not Magazines?
From the calculation of the region \( (T ∩ R) \setminus (T ∩ R ∩ M) \), this is:
\[
5
\]

### d. How many use Television and Magazines but not Radio?
From the calculation of the region \( (T ∩ M) \setminus (T ∩ R ∩ M) \), this is:
\[
15
\]

---

Would you like me to draw the Venn diagram or go deeper into any specific part of the explanation?

### Related Questions:
1. How can you use the inclusion-exclusion principle to solve overlapping set problems?
2. What is the total number of clients who use exactly two services?
3. How many clients do not use any of the services?
4. What is the formula to find the number of people using at least one service?
5. How would the solution change if there were an additional service (e.g., Online)?

### Tip:
In Venn diagram problems, always start from the most specific intersection (all sets overlapping) and work outward to avoid double-counting.

An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. Find: a. how many use only Radio? b. how many use only Television? c. how many use Radio and Television but not Magazine? d. how many use Television and Magazine but not Radio?

Solve a set theory problem involving 170 clients of an advertising agency using Television, Radio, and Magazines. Learn how to calculate the number of clients using only one or two services with the inclusion-exclusion principle and Venn diagrams.

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