GMAT Club World Cup 2022 (DAY 4): Messi plans a trip to Qatar and

Question

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

ThatDudeKnows · Answer

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.

$!vakumar.m · Answer

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

Kinshook · Answer

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

mainbhiankit · Answer

I solved this question using case by case approach. If anyone has a shorter approach, please feel free to tag me.

Assuming that Messi will take 9 friends (all 5 footballers): 5C5*5C4 = 5
Assuming that Messi will take 9 friends (but 4 footballers): 5C4*5C5 = 5
Therefore for 9 friends, the number of teams possible = 5+5 = 10

If you calculate this up to 4 friends (since he will take at least 4 footballer friends), the number of possible selections is 191.

gmatophobia · Answer

We know that at max Messi can invite 9 of his friends, however he wants to invite at least 4 football players. so the number of players that can be invited will lie between 4 to 9, both inclusive
 
Number of ways Messi can invite 4 players = 5C4 = 5
Number of ways Messi can invite 5 players = 5C4*5C1 + 5C5 = 26
Number of ways Messi can invite 6 players = 5C4*5C2 + 5C5*5C1 = 55
Number of ways Messi can invite 7 players = 5C4*5C3 + 5C5*5C2 = 60
Number of ways Messi can invite 8 players = 5C4*5C4 + 5C5*5C3 = 35
Number of ways Messi can invite 9 players = 5C4*5C5 + 5C5*5C4 = 10

If we sum up the above we obtain a total of 191

IMO C

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

GMAT Club World Cup 2022 (DAY 4): Messi plans a trip to Qatar and

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

GMAT Club World Cup 2022 (DAY 4): Messi plans a trip to Qatar and

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards