Define variables for the number of students in each region of the Venn diagram.

Step 1: Define variables for the number of students in each region of the Venn diagram. Let V, T, and H represent the sets of students who played Volleyball, Tennis, and Hockey, respectively. Let V_o be the number of students who played Volleyball only. Let T_o be the number of students who played Tennis only. Let H_o be the number of students who played Hockey only. Let VT_o be the number of students who played Volleyball and Tennis only (not Hockey). Let VH_o be the number of students who played Volleyball and Hockey only (not Tennis). Let TH_o be the number of students who played Tennis and Hockey only (not Volleyball). Let VTH be the number of students who played all three games (|V T H|). This is the value we need to find. Let N_0 be the number of students who played none of the games. Step 2: Translate the given information into equations. We are given: • Total students = 122 • |V| = 59 • |T| = 58 • |H| = 53 • V_o = 28 • |V T| = 22 VT_o + VTH = 22 • |T H| = 27 TH_o + VTH = 27 • N_0 = VTH (Those who did not play any games were equal to those who played all three games). Step 3: Express unknown regions in terms of VTH. From the intersection data: VT_o = 22 - VTH TH_o = 27 - VTH Now use the total for each sport: For Volleyball: |V| = V_o + VT_o + VH_o + VTH = 59 Substitute V_o = 28 and VT_o = 22 - VTH: 28 + (22 - VTH) + VH_o + VTH = 59 50 + VH_o = 59 VH_o = 59 - 50 = 9 For Hockey: |H| = H_o + VH_o + TH_o + VTH = 53 Substitute VH_o = 9 and TH_o = 27 - VTH: H_o + 9 + (27 - VTH) + VTH = 53 H_o + 36 = 53 H_o = 53 - 36 = 17 For Tennis: |T| = T_o + VT_o + TH_o + VTH = 58 Substitute VT_o = 22 - VTH and TH_o = 27 - VTH: T_o + (22 - VTH) + (27 - VTH) + VTH = 58 T_o + 49 - VTH = 58 T_o = 58 - 49 + VTH = 9 + VTH Step 4: Use the total number of students to find VTH. The sum of all distinct regions must equal the total number of students: Total students = V_o + T_o + H_o + VT_o + VH_o + TH_o + VTH + N_0 Substitute all expressions in terms of VTH and N_0 = VTH: 122 = 28 + (9 + VTH) + 17 + (22 - VTH) + 9 + (27 - VTH) + VTH + VTH Combine the constant terms: 28 + 9 + 17 + 22 + 9 + 27 = 112 Combine the VTH terms: VTH - VTH - VTH + VTH + VTH = VTH So the equation becomes: 122 = 112 + VTH VTH = 122 - 112 VTH = 10 Step 5: State the final advice. The number of students who played all three games is VTH = 10. The bursary is awarded to students who can play all three games. The coach should forward 10 students for consideration of the bursary. Send me the next one 📸