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14 Jul 2022, 08:00
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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
A. 63
B. 64
C. 191
D. 192
E. 1023
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Updated on: 15 Jul 2022, 08:08
Originally posted by ThatDudeKnows on 14 Jul 2022, 08:08.
Last edited by ThatDudeKnows on 15 Jul 2022, 08:08, edited 1 time in total.
Last edited by ThatDudeKnows on 15 Jul 2022, 08:08, edited 1 time in total.
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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
A. 63
B. 64
C. 191
D. 192
E. 1023
|
How many ways are there for him to invite the NON-football players? There are five of them and each could either be in or out. That means there are 2^5 = 32 ways.
How many ways are there for him to invite the FOOTBALL players? Well, he could either invite all five OR invite four of the five.
There is one way to invite all five. There are five ways to invite four of the five.
So, there are 1*32 ways to invite all five football players plus some combination of non-football players. That's 32. But we need to subtract the 1 way of inviting all 10 since he invites at most 9. So, 31.
And there are 5*32 ways to invite four of the five football players plus some combination of non-football players. That's 160.
31+160 = 191
Answer choice C.
14 Jul 2022, 08:19
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atleast 4 friends to max 9 friends can accompany
we have 5 football playing friends and 5 non football playing friends
Messi has an option of taking 4,5,6,7,8 or 9 friends
and 4/5 can be selected from football playing friends
and 1,2,3,4,5 can be selected from non football playing friends
group of 4 : 5C4 = 5
group of 5 : 5C4*5C1 + 5C5 = 26
group of 6: 5C4*5C2 + 5C5*5C1 = 55
group of 7: 5C4*5C3 + 5C5*5C2 = 60
group of 8: 5C4*5C4 + 5C5*5C3 = 35
group of 9: 5C4*5C5 + 5C5*5C4 = 10
therefore total no of groups = 191
Answer: C
we have 5 football playing friends and 5 non football playing friends
Messi has an option of taking 4,5,6,7,8 or 9 friends
and 4/5 can be selected from football playing friends
and 1,2,3,4,5 can be selected from non football playing friends
group of 4 : 5C4 = 5
group of 5 : 5C4*5C1 + 5C5 = 26
group of 6: 5C4*5C2 + 5C5*5C1 = 55
group of 7: 5C4*5C3 + 5C5*5C2 = 60
group of 8: 5C4*5C4 + 5C5*5C3 = 35
group of 9: 5C4*5C5 + 5C5*5C4 = 10
therefore total no of groups = 191
Answer: C
