\( \square 0 / 5 \) pts \( 2 \rightleftarrows 19 \) Details A survey asked 225 people what alternative transportation modes they use. The results are below. 98 walk 117 use the bus 110 ride a bicycle 44 walk and use the bus 39 walk and ride a bicycle 62 ride the bus and ride a bicyle 21 said they use all three modes of pransportation How many people only ride the bus? How many people don't use any alternate transportation? Question Help: Video 1 Video 2 Submit Question

To solve the problem, we can use the principle of inclusion-exclusion and a Venn diagram approach. Let's define the sets: - Let \( W \) be the set of people who walk. - Let \( B \) be the set of people who use the bus. - Let \( C \) be the set of people who ride a bicycle. From the information given, we have: - \( |W| = 98 \) - \( |B| = 117 \) - \( |C| = 110 \) - \( |W \cap B| = 44 \) (people who walk and use the bus) - \( |W \cap C| = 39 \) (people who walk and ride a bicycle) - \( |B \cap C| = 62 \) (people who use the bus and ride a bicycle) - \( |W \cap B \cap C| = 21 \) (people who use all three modes) ### Step 1: Calculate the number of people in each category Using the inclusion-exclusion principle, we can find the number of people who use at least one mode of transportation: \[ |W \cup B \cup C| = |W| + |B| + |C| - |W \cap B| - |W \cap C| - |B \cap C| + |W \cap B \cap C| \] Substituting the values: \[ |W \cup B \cup C| = 98 + 117 + 110 - 44 - 39 - 62 + 21 \] Calculating this step-by-step: 1. \( 98 + 117 + 110 = 325 \) 2. \( 44 + 39 + 62 = 145 \) 3. \( 325 - 145 + 21 = 201 \) So, \( |W \cup B \cup C| = 201 \). ### Step 2: Calculate the number of people who don't use any alternate transportation The total number of people surveyed is 225. Therefore, the number of people who don't use any alternate transportation is: \[ 225 - |W \cup B \cup C| = 225 - 201 = 24 \] ### Step 3: Calculate the number of people who only ride the bus To find the number of people who only ride the bus, we can use the following formula: \[ |B \text{ only}| = |B| - (|W \cap B| + |B \cap C| - |W \cap B \cap C|) \] Substituting the values: \[ |B \text{ only}| = 117 - (44 + 62 - 21) \] Calculating this step-by-step: 1. \( 44 + 62 = 106 \) 2. \( 106 - 21 = 85 \) 3. \( 117 - 85 = 32 \) So, the number of people who only ride the bus is \( 32 \). ### Summary of Results - The number of people who only ride the bus: **32** - The number of people who don't use any alternate transportation: **24**