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705-805 (Hard)|   Math Related|         
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 11:50
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Total Outcomes: 8C3 = 56

Possible Outcomes:
case 1) none selected from A,B,C,D. three selected from amongst E,F,G,H. 4C3 = 4
case 2) one selected from either A,B or C,D and rest two selected from amongst E,F,G,H. 2C1*4C2*2=24
case 3) one selected each from A,B and C,D and rest one selected from amongst E,F,G,H. 2C1*2C1*4C1=16
Total possible outcomes: 4+24+16=44

Probability = 44/56 = 11/14. X=11, Y=14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 12:06
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56 would be overall Y, use combinations to arrive at this, 8!/3!*5!, now look for combinations where they all will not have these incompatible people, there are total 6 of them each, if u place two of them one of other 6 people will have the 3rd post. Ensure you do double counting for other hence total 56-12 which is 44. Now 44/56 will be 11/14. Which is X and Y respectively.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 12:09
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its ofcourse 1-12/56
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 12:12
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Total outcome, 8C3
=8!/3!*5! = 8*7 = 56

Total unfavourable outcome = Total invalid committes = Committees with AB pair + Committes with CD Pair = 2* (1*6) = 12

Probability = (Total- unfav)/ Total = 56-12/56 = 44/56 = 11/14

So X = 11, Y = 14.
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 12:18
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Possible valid outcomes-
2C1*2C1*4C1 + (2C1*4C2)2 + 4C3
Outcome 1- 1 from A&B and 1 from C&D and 1 from rest 4.
Outcome 2- 1 from A&B and 2 from rest 4
Outcome 3- 1 from B&C and 2 from rest 4
Outcome 4- 3 from rest 4.
Total outcome- 8C3.
Valid/Total= 11/14
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 12:20
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Total ways of forming a committee are 8C3 = 8*7

Unfavourable cases are = AB with all other 6 members or CD with all other 6 members = 12 ways of forming an invalid committee

Thus favourable cases are = 1 - (12/8*7) = 11/14

Answer = X is 11 and Y is 14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 12:38
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The answer is 11/14. I made 4 cases:
(1) 3 people from E, F, G, H- 4C3
(2) 2 people from E, F, G, H and 1 from C, D - 4C2*2C1
(3) 2 people from E, F, G, H and 1 from A, B - 4C2*2C1
(4) 1 from E, F, G and H, 1 from C, D and 1 from A, B- 4C1*2C1*2C1

Total Ways = 8C3
Answer = 11/14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 12:56
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There can be 4 cases here

Case 1: pick 3 committes excluding the incompatible pairs i.e from E F G H i.e 4C3 which is 4 ways

Case 2: One from one pair, one from the other pair and 1 from remaining

Eg From A, B , C , D , E , F , H ----> Pick one from AB, one from CD and 2 from E F G H = 2 x 2 x 4 = 16 ways

Case 3: Pick one from A B , and 2 from E F G H i.e 2 x 4C2 = 12 ways

Case 4: Pick one from C, D and 2 from E F G H i.e Again 2 x 4C2 = 12 ways

Total = 4 + 16 + 12 + 12 = 44

Sample size is 8C3 which is 56

Hence prob is 44/56 i.e 11/14


Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 14:08
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Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
Show more

Total number of ways 3 people can be chosen from 8 people = 8C3
The committee is invalid when either A or B are together, or either C or D are together. When either of the pair exists, the third person is chosen from the six remaining.

Number of ways of choosing 1 member from 6 available members = 6

Invalid pairs = 6 + 6 = 12

Valid pairs = 8C3 - 12 = 56 - 12 = 44

Fraction = 44/56

= 22/28

= 11 / 14

X = 11 ; Y = 14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 14:43
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To calculate probablity, we need
1. Total Outcomes
2. Favorable outcomes - where Aly and Ben & Cam & Dan are not together.

1. 8C3: we are selecting 3 people from 8 so the combinations formed are 56.

2. To find outcomes where A&B and C&D are not together, we can calculate where they are together and subtract that from the total outcome.

so when A&B are together: positions are as follows- A, B, _. -------> this blank can be filled in 6 ways.
and similarly when A & B are together: positions are as follows- C, D, _. ---------this blank can be filled in 6 ways too.

So finally there 6+6= 12 ways where the pairs are together.

56-12=44 which is our favourable outcome and

X/Y = 44/56
= 11/14
so X= 11
and Y = 14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 17:22
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subtract total Invalid committees from the valid committees
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 17:28
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Total combinations(Y)=8c3=56
Bad combination(A,B or C,D together will have 1 seat letf with 6 candidates)=6c1 for each A,B or C,D

valid selection(X)=8c3-2(6c1)=56-2(6)=44
X/Y=44/56=11/14

Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 17:56
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Let us denote everyone by their first letter alone (A, B, C, and so on).
(A,B) and (C,D) cannot be present in the three member group. Since this is a three member group, both pairs cannot occur at the same time.

All possible combinations = 8C3=56
Invalid combinations where (A,B) is present=6C1=6
Invalid combinations where (C,D) is present=6C1=6
Valid combinations=All possible combinations-Invalid combinations=56-6-6=44

Probability that a valid committee is formed=Valid combinations/All possible combinations=44/56=11/14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 18:41
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no. of valid cases:
3 people from E,F,G,H => 4 ways
If any one of the member is selected from ABCD =>
If A then rest of 2 people get selected from rest of the people (c/d, e,f,g,h) total 5. so 5c2 that would be 10 ways.
Similarly for rest of others BCD. then it would be total 40 ways.
Not total would come 44 ways/8C3 => 44x3x2/(8x7x6) => 11/14
X=11 Y=14
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 19:18
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X = Valid committees, Y = Total possible committee combinations if no conflicts of interest existed

Y = 8C3 = 56
X = Total - Invalid committees.

Total invalid committees = No. of committees where Aly & Ben are together + No. of committees where Cam and Dan are together.
If any 2 members are already known for a selection of 3 people committee, we just need to choose one more person from the rest of the group. e.g., if Aly and Ben are going, Committee = (A, B, one of {C, D, E, F, G, H}) = 6C1 = 6. Similarly, if Cam and Dan both get selected, 6 possible committees can be made. Total invalid = 12

X = 56 - 12 = 44

X/Y = 44/56 = 11/14
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 20:02
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x=11 y=14 because 44/56 simply dividing both for 4 so we got 11/14 -> x=11 y=14
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
Show more
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 20:31
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Total ways of forming committee = 8c3 = 56 . When AB and CD form committee they become invalid committee we need to find the cases where we form valid committee so to find the number of valid cases= total cases - invalid cases . How is invalid committee AB and CD formed - AB(3rd person) - A and B can be selected in 1 way each and 3rd person can be selected in 6c1 = 6 ways . Similarly for CD we can form CD(3RD PERSON) in 6 ways . Total ways to form invalid committee = 12 (6+6) .No of valid cases will be = 56-12 = 44 cases . X/Y= VALID CASES /TOTAL OUTCOME =44/56 =11/14
Bunuel
A company is forming a three-person leadership committee from a pool of eight qualified candidates: Aly, Ben, Cam, Dan, Emi, Fay, Gia, and Hal. Due to conflicts of interest, two pairs of candidates are incompatible: Aly and Ben cannot both serve on the committee together, and Cam and Dan cannot both serve on the committee together. A committee is considered valid only if its three members do not include either of these pairs of incompatible candidates.

Consider the following incomplete sentence:

If three candidates are chosen at random, the probability that the selected candidates form a valid committee is X / Y , where the fraction is expressed in simplest form.

Select for X and Y values that are consistent with the information provided. Make only two selections, one in each column.
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 20:46
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One getting selected from each player and one from remaining three + one from either of the players and 2 from remaining player + all three from remaining four player

Answer is 11/14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 21:06
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Total numer of groups with 8 candidates and 3-person groups: 8C3 = 56
From these, there are 6 groups that contain Aly and Ben and another 6 groups that contain Cam and Dan: 6+6=12
Valid groups: 56-12=44

44/56 = 11/14

X=11
Y=14
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GMAT Club World Cup 2026 (Day 1): A company is forming a three-person 06 Jul 2026, 21:09
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Total Cases = 8C3 =56 cases
AB together = 6 cases
Similarly CD together = 6 Cases
Valid Cases = 56-6-6=44
P(Valid) = 44/56 =0.786 11/16
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