Bunuel wrote:
Messi plans a trip to Qatar World Cup 2022 and intends to invite at most 9 of his 10 friends with him. 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 ?
A. 63
B. 64
C. 191
D. 192
E. 1023
Total friends 10
Football friends - 5
Non-Football friends - 5
So Messi wants to take at least 4 of his 5 football friends and at most a total of 9 friends
Let's take the first option where Messi takes 4 of his 5 football friends then we have the following scenarios
Messi takes 0 Non-Football friends = 5C4 * 1
OR
Messi takes 1 Non-Football friends = 5C4 * 5C1
OR
Messi takes 2 Non-Football friends = 5C4 * 5C2
OR
Messi takes 3 Non-Football friends = 5C4 * 5C3
OR
Messi takes 4 Non-Football friends = 5C4 * 5C4
OR
Messi takes 5 Non-Football friends = 5C4 * 5C5
Total combinations are = = 5C4 + 5C4 * 5C1 + 5C4 * 5C2 + 5C4 * 5C3 + 5C4 * 5C4 + 5C4 * 5C5 = 5C4(1 + 5 + 10 + 10 + 5 + 1) = 160
Similarly, let's take the option where Messi takes all 5 of his 5 football friends. Then we have the following scenarios
Total combinations are = = 5C5 + 5C5 * 5C1 + 5C5 * 5C2 + 5C5 * 5C3 + 5C5 * 5C4 = 5C5(1 + 5 + 10 + 10 + 5) = 31
Total combinations = 160 + 31 = 191
IMHO Option C