Given: Messi plans a trip to Qatar World Cup 2022 and intends to invite at most 9 of his 10 friends with him.
Asked: If half of his 10 friends are football players and Messi wants to invite at least 4 football players to Qatar, then how many different groups of friends can Messi form for the trip ?
Case 1: Group of friends for the trip include 4 football players
Number of ways to select 4 football players out of 5 = 5C4 = 5
Number of ways to select at most 5 other friends = 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 1 + 5 + 10 + 10 + 5 + 1 = 32
Total number of ways to select friends for the trip including 4 football players = 5*32 = 160
Case 2: Group of friends for the trip include all 5 football players
Number of ways to select 5 football players out of 5 = 5C5 = 1
Number of ways to select at most 4 other friends = 5C0 + 5C1 + 5C2 + 5C3 + 5C4 = 1 + 5 + 10 + 10 + 5 = 31
Total number of ways to select friends for the trip including 5 football players = 31
Total number of group of friends for the trip = 160 + 31 = 191
IMO C
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Kinshook Chaturvedi
Email: kinshook.chaturvedi@gmail.com