This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Morning rubangakeneakena — let's get this done. Here's the solution to the problem: Let H be the set of students who play hockey, F be the set of students who play football, and V be the set of students who play volleyball. Given information: Total students, N = 87 n(H) = 40 n(F) = 42 n(V) = 47 n(H V) = 15 n(V F) = 17 n(H F) = 21 Each student plays at least one game, so n(H F V) = 87. 1. Determine the number of students who play all three games. Step 1: Use the Principle of Inclusion-Exclusion for three sets. n(H F V) = n(H) + n(F) + n(V) - n(H F) - n(H V) - n(F V) + n(H F V) Step 2: Substitute the given values into the formula. 87 = 40 + 42 + 47 - 21 - 15 - 17 + n(H F V) Step 3: Calculate the sum and differences. 87 = 129 - 53 + n(H F V) 87 = 76 + n(H F V) Step 4: Solve for n(H F V). n(H F V) = 87 - 76 n(H F V) = 11 So, 11 students play all three games. 2. Using a Venn diagram, represent the information. Let x = n(H F V) = 11. The number of students playing exactly two games are: • Only Hockey and Football: n(H F) - x = 21 - 11 = 10 • Only Hockey and Volleyball: n(H V) - x = 15 - 11 = 4 • Only Football and Volleyball: n(F V) - x = 17 - 11 = 6 The number of students playing exactly one game are: • Only Hockey: n(H) - (10 + 4 + 11) = 40 - 25 = 15 • Only Football: n(F) - (10 + 6 + 11) = 42 - 27 = 15 • Only Volleyball: n(V) - (4 + 6 + 11) = 47 - 21 = 26 A Venn diagram would show three overlapping circles for H, F, and V. The regions would contain the following numbers: • Center (H F V): 11 • H only F: 10 • H only V: 4 • F only V: 6 • H only: 15 • F only: 15 • V only: 26 3. How many students play only 2 of the games? Step 1: Sum the number of students playing exactly two games. n(only 2 games) = n(H F only) + n(H V only) + n(F V only) Step 2: Substitute the calculated values. n(only 2 games) = 10 + 4 + 6 n(only 2 games) = 20 So, 20 students play only 2 of the games. 4. If a student is picked at random from the class, find the probability that the student plays only one of the games. Step 1: Calculate the total number of students who play only one game. n(only 1 game) = n(H only) + n(F only) + n(V only) Step 2: Substitute the calculated values. n(only 1 game) = 15 + 15 + 26 n(only 1 game) = 56 Step 3: Calculate the probability. P(only 1 game) = Number of students playing only 1 gameTotal number of students P(only 1 game) = (56)/(87) The probability is (56)/(87). Send me the next one 📸