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
Step 1: Define sets and given information. Let A be the set of attackers, D be the set of defenders, and M be the set of midfielders. Total players = 20. Given: |A| = 8 |D| = 12 |M| = 12 |A M| = 5 (attack and midfield) |D M| = 6 (defense and midfield) |A (M D)| = 2 (only attack) |A D M| = 3 (all three roles) Step 2: Calculate the number of players in each distinct region of the Venn diagram. • Players in all three roles (A D M): 3 • Players in attack and midfield only (A M D): |A M| - |A D M| = 5 - 3 = 2 • Players in defense and midfield only (D M A): |D M| - |A D M| = 6 - 3 = 3 • Players in attack only (A (M D)): 2 • Players in attack and defense only (A D M): The total in A is 8. We have 2 (A only), 2 (A and M only), 3 (all three). So, |A D M| = |A| - (|A (M D)| + |A M D| + |A D M|) |A D M| = 8 - (2 + 2 + 3) = 8 - 7 = 1 • Players in defense only (D (A M)): The total in D is 12. We have 1 (A and D only), 3 (D and M only), 3 (all three). So, |D (A M)| = |D| - (|A D M| + |D M A| + |A D M|) |D (A M)| = 12 - (1 + 3 + 3) = 12 - 7 = 5 • Players in midfield only (M (A D)): The total in M is 12. We have 2 (A and M only), 3 (D and M only), 3 (all three). So, |M (A D)| = |M| - (|A M D| + |D M A| + |A D M|) |M (A D)| = 12 - (2 + 3 + 3) = 12 - 8 = 4 Step 3: Verify the total number of players. Sum of all distinct regions: 2 (A only) + 5 (D only) + 4 (M only) + 1 (A and D only) + 2 (A and M only) + 3 (D and M only) + 3 (A and D and M) = 2 + 5 + 4 + 1 + 2 + 3 + 3 = 20. This matches the total number of players invited. i. Illustrate the information in a Venn diagram A Venn diagram would show three overlapping circles (A, D, M) with the following numbers in each distinct region: • Only Attack (A only): 2 • Only Defense (D only): 5 • Only Midfield (M only): 4 • Attack and Defense only (A D M): 1 • Attack and Midfield only (A M D): 2 • Defense and Midfield only (D M A): 3 • Attack, Defense, and Midfield (A D M): 3 ii. Find the number of players that can play: a) only midfield This is the number of players in the "Midfield only" region. Number of players = 4 The number of players that can play only midfield is 4. b) exactly 2 roles This is the sum of players in the regions where exactly two sets overlap: (Attack and Defense only) + (Attack and Midfield only) + (Defense and Midfield only) Number of players = 1 + 2 + 3 = 6 The number of players that can play exactly 2 roles is 6. Send me the next one 📸