Question 10 A survey asked buyers whether color, size, or brand influenced their choice of cell phone. The results are below. 141 said size. 147 said brand. 269 said color. 51 said size and brand. 112 said color and size. 115 said color and brand. 33 said all three. 71 said none of these How many buyers were influenced by color and size, but not brand? \( \square \) How many buyers were not influenced by color? How many buyers were surveyed? Question Help: Video Submit Question

To solve the problem, we can use the principle of inclusion-exclusion and Venn diagrams. Let's define the following: - Let \( C \) be the set of buyers influenced by color. - Let \( S \) be the set of buyers influenced by size. - Let \( B \) be the set of buyers influenced by brand. From the survey results, we have: - \( |C| = 269 \) - \( |S| = 141 \) - \( |B| = 147 \) - \( |S \cap B| = 51 \) - \( |C \cap S| = 112 \) - \( |C \cap B| = 115 \) - \( |C \cap S \cap B| = 33 \) - \( |None| = 71 \) ### 1. How many buyers were 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: \[ |C \cap S \cap \overline{B}| = |C \cap S| - |C \cap S \cap B| \] Substituting the values: \[ |C \cap S \cap \overline{B}| = 112 - 33 = 79 \] So, **79 buyers were influenced by color and size, but not brand.** ### 2. How many buyers were not influenced by color? To find the number of buyers not influenced by color, we first need to find the total number of buyers surveyed. We can use the inclusion-exclusion principle to find the total number of buyers influenced by at least one of the factors (color, size, brand). The formula for the union of three sets is: \[ |C \cup S \cup B| = |C| + |S| + |B| - |C \cap S| - |C \cap B| - |S \cap B| + |C \cap S \cap B| \] Substituting the values: \[ |C \cup S \cup B| = 269 + 141 + 147 - 112 - 115 - 51 + 33 \] Calculating step-by-step: 1. \( 269 + 141 + 147 = 557 \) 2. \( 557 - 112 = 445 \) 3. \( 445 - 115 = 330 \) 4. \( 330 - 51 = 279 \) 5. \( 279 + 33 = 312 \) So, \( |C \cup S \cup B| = 312 \). Now, to find the total number of buyers surveyed, we add those who were influenced by at least one factor to those who were not influenced by any: \[ \text{Total Buyers} = |C \cup S \cup B| + |None| = 312 + 71 = 383 \] Now, to find the number of buyers not influenced by color: \[ | \overline{C} | = \text{Total Buyers} - |C| = 383 - 269 = 114 \] So, **114 buyers were not influenced by color.** ### 3. How many buyers were surveyed? From our calculations above, the total number of buyers surveyed is **383**. ### Summary of Answers: 1. Buyers influenced by color and size, but not brand: **79** 2. Buyers not influenced by color: **114** 3. Total buyers surveyed: **383**