A survey asked buyers whether color, size, or brand influenced their choice of cell phone. The
results are below.
185 said size.
147 said brand.
206 said color.
99 said size and brand.
111 said color and size.
52 said color and brand.
39 said all three.
76 said none of these
How many buyers were influenced by color and size, but not brand?
How many buyers were not influenced by color?
How many buyers were surveyed?
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UpStudy AI · Accepted Answer

To solve the problem, we will use the principle of inclusion-exclusion and Venn diagrams to analyze the data provided.

### Known Conditions:
- Total respondents influenced by size: $$ S = 185 $$
- Total respondents influenced by brand: $$ B = 147 $$
- Total respondents influenced by color: $$ C = 206 $$
- Respondents influenced by both size and brand: $$ S \cap B = 99 $$
- Respondents influenced by both color and size: $$ C \cap S = 111 $$
- Respondents influenced by both color and brand: $$ C \cap B = 52 $$
- Respondents influenced by all three factors: $$ S \cap B \cap C = 39 $$
- Respondents influenced by none of these factors: $$ N = 76 $$

### Step 1: Find the number of buyers influenced by color and size, but not brand.
To find the number of buyers influenced by color and size but not brand, we can use the formula:

$$
(C \cap S) - (S \cap B \cap C)
$$

Substituting the known values:

$$
(C \cap S) - (S \cap B \cap C) = 111 - 39
$$

### Step 2: Find the number of buyers not influenced by color.
To find the number of buyers not influenced by color, we can use the total number of buyers surveyed and subtract those influenced by color.

### Step 3: Find the total number of buyers surveyed.
The total number of buyers surveyed can be calculated as:

$$
\text{Total} = (S + B + C) - (S \cap B + C \cap S + C \cap B) + (S \cap B \cap C) + N
$$

Now, let's calculate these values step by step.

First, we will calculate the number of buyers influenced by color and size but not brand. Then, we will calculate the total number of buyers surveyed. Finally, we will find the number of buyers not influenced by color.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$111-39$$
- step1: Subtract the numbers:
  $$72$$
Calculate or simplify the expression $$ (185 + 147 + 206) - (99 + 111 + 52) + 39 + 76 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(185+147+206\right)-\left(99+111+52\right)+39+76$$
- step1: Add the numbers:
  $$538-\left(99+111+52\right)+39+76$$
- step2: Add the numbers:
  $$538-262+39+76$$
- step3: Calculate:
  $$391$$
### Step 1: Buyers influenced by color and size, but not brand
The number of buyers influenced by color and size but not brand is:

$$
111 - 39 = 72
$$

### Step 2: Total number of buyers surveyed
The total number of buyers surveyed is:

$$
(185 + 147 + 206) - (99 + 111 + 52) + 39 + 76 = 391
$$

### Step 3: Buyers not influenced by color
To find the number of buyers not influenced by color, we subtract those influenced by color from the total number of buyers surveyed:

$$
391 - 206 = 185
$$

### Summary of Results:
1. The number of buyers influenced by color and size, but not brand: **72**
2. The number of buyers not influenced by color: **185**
3. The total number of buyers surveyed: **391**
- **72** buyers were influenced by color and size, but not brand. - **185** buyers were not influenced by color. - **391** buyers were surveyed in total.

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Solución

Known Conditions:

Step 1: Find the number of buyers influenced by color and size, but not brand.

Step 2: Find the number of buyers not influenced by color.

Step 3: Find the total number of buyers surveyed.

Step 1: Buyers influenced by color and size, but not brand

Step 2: Total number of buyers surveyed

Step 3: Buyers not influenced by color

Summary of Results:

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