Eight persons in an organization including Andrew and Bob were to be d

Question

Competition Mode Question

Eight persons in an organization including Andrew and Bob were to be divided in two groups of 4 members each. The total number of groups containing both Andrew and Bob is what fraction of the total number of groups which can be formed?

A. 1/4
B. 1/70
C. 3/14
D. 1/10
E. 11/14

CareerGeek · Accepted Answer

Number of ways in which 2 groups of 4 that can be formed =  8c_4  =  \frac{8*7*6*5}{4!} = 70

Favorable outcome = Andrew & Bob to be in same team of 4
--> Favorable ways = selecting 2 members from remaining 6 =  6c_2 = 6*5/2! = 15

--> Required Probability =  \frac{15}{70}  =  \frac{3}{14}

Option C

Archit3110 · Answer

total possible groups l 8c4 ; 70
group containing andrew and bob ; 6c2 ;
P ; 15/70 ; 3/14
IMO C

Eight persons in an organization including Andrew and Bob were to be divided in two groups of 4 members each. The total number of groups containing both Andrew and Bob is what fraction of the total number of groups which can be formed?

SirFrik · Answer

There are 8C4 ways to choose the first of the two groups and only 1 way to choose the second group after the first is chosen so: 140 ways to create 2 groups of 4 members each.

Andrew and Bob can be both in the first group or both in the second group.

If both are in the first group, we anchor them in and choose 2 people from the remaining 6 to complete the first group. This can be done in 15 ways. The other scenario, that is when they are both in the second group, is the mirror image of the former scenario. So there are 15+15=30 ways for this to happen.

30/140 or 3/14.

Answer is C

freedom128 · Answer

The total number of groups which can be formed = 8C4 = 70 groups (this includes the second group and sequence is not important here)

The total number of groups containing both Andrew and Bob (A, B, _, _) = 6C2 = 15

Fraction of total groups = 15/70 = 3/14

FINAL ANSWER IS (C)

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eakabuah · Answer

Total number of employees = 8; 
To form a group of 4 employees that contain Andrew and Bob out of 8, we only need to add 2 of the remaining 6 employees to Andrew and Bob. This can be achieved in 6C2 ways = 15 ways.

The total number of possible ways to form groups of 4 out of 8 = 8C4 = 70 ways.

The number of groups containing Andrew and Bob as a fraction of the total possible groups = 15/70 = 3/14.

The answer is C.

SREYNANS · Answer

Im confused

by using formula by theory given by  chetan2u   https://gmatclub.com/forum/topic215915.html#p1667366 
 Formula- 
 If you divide 'xa' items in 'x' groups of 'a' items each, the number of ways=  \frac{(xa)!}{x!(a!)^x}

so total cases are 35, and no of groups in which A and B are together is 6C2 =15
my answer is 15/35.

Can someone clarify.
Regards
Sreynans

SREYNANS · Answer

Ok got it, the no.of groups would be double the no. of ways. so 15/70 =3/14 
Ans C

ScottTargetTestPrep · Answer

The total number of 4-person groups is:

8C4 * 4C4 =(8 x 7 x 6 x 5) / (4 x 3 x 2) * 1 = 70

Now let’s determine the total number of 4-person groups containing both Andrew and Bob. 
Since Bob and Andrew are guaranteed to be chosen (2C2 = 1), we have 6 remaining individuals to fill the 2 remaining slots:

2C2 * 6C2 = 1 * (6 x 5)/2 = 15

Therefore, the probability is 15/70 = 3/14.

Answer: C

ConfusedSEA · Answer

Question:  The total number of groups containing both Andrew and Bob  is what fraction of the  total number of groups which can be formed

Bolded 1: The total number of groups containing both Andrew and Bob= 8C4= 70

Bolded 2: total number of groups which can be formed = 6C2=15
6 represents the remaining employees after removing Andrew & Bob
2 represents the vacancies left from an initial 4 which was taken by Andrew & Bob

Answer= 15/70=3/14  (C)

Tiburcio1987 · Answer

Can you please explain the logic of how you get 6C2?

Archit3110 · Answer

It's a given condition that A&B both have to be part of the selection so in that case we combine them say X which can be chosen in 2c2 ways
then we are left with 6 members & 2 members are to be chosen. 6c2
2c2:1 
1*6c2 
 Tiburcio1987

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Akshat1994 · Answer

Can anyone please explain why the total no: of cases are 8C4 and not 8C4/2

Say A, B, C, D, E, F , G, H are 8 persons

Two groups of ABCD and EFGH will be same as EFGH and ABCD.

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GMATGuruNY · Answer

Since the 8 people are divided into two groups of 4, Andrew must be selected for one of the two groups.
If we select one of the two groups:
Since Andrew has an equal chance of being included in either group, P(Andrew is included in the selected group) = 1/2
Since there are 3 spots left in Andrew's group -- and 7 remaining people who could occupy these 3 spots -- P(Bob is one of the 3 people selected to join Andrew) = 3/7
Thus:
P(Andrew and Bob are in the same group) = 1/2 * 3/7 = 3/14

Since the probability that Andrew and Bob are in the same group = 3/14, they must be included together in 3/14 of all possible 4-member groups.

C

GMATGuruNY · Answer

Case 1:
 How many pairs can be formed from four people A, B, C and D? 
Answer: 4C2  = \frac{4*3}{2*1} = 6 
The approach above counts the following options:
AB, AC, AD, BC, BD, CD

Case 2:
 How many ways can four people be divided into two pairs? 
Answer:  \frac{4C2}{2!} = \frac{6}{2} = 3 
The approach above counts the following options:
AB and CD
AC and BD
AD and BC

How many groups of 4 can be formed from 8 people? 
This question is similar to Case 1 above.
Thus:
Number of possible 4-person groups = 8C4

JhonnyMe · Answer

Hey everyone, appreciate all the solutions and approaches provided above. However, I had a slight different interpretation of the question.
A & B have to be together in any of the groups. So if A & B are in group 1, and not in group 2 then case satisfied. Or if A & B are in group 2 and not in group 1, then also case is satisfied.

So for case 1 (Where A&B in group 1): 6C2
And for case 2 (Where A&B in group 2): 6C4 
I am coming to this conclusion by focusing on group 1 alone. And if A&B are not in group 1 in case 2, then they are together in group 2.

Case 1+ Case 2 = 30
Total possibilities: 8C4 = 70.

This gives me a probability of 3/7.

Ik this is the wrong answer. Would love if anyone can help me understand where I am going wrong!

JhonnyMe · Answer

Hey SirFrik,

my thought process went similar to what you have provided here. My question here is how did you arrive to 140 ways? There are 8C4 ways to choose group 1 which is 70 ways. The remaining people not selected for group 1 automatically fall into group 2. Why did you multiply by 2 to arrive to 140? Or did you have an alternate approach?

culpaquam · Answer

The groups are not labeled. 
Whether we get 1,2,3,4 and 5,6,7,8 or 5,6,7,8 and 1,2,3,4 - we don't care, it is the same result. 
So the number of combinations should be divided by 2 to exclude the orderings that are equivalent. 
Thus there are 35 groups in total, so the final fraction is indeed 15/35 = 3/7

Bunuel   KarishmaB  what's your opinion?

KarishmaB · Answer

The wording of the question is suspect. It first talks about "... 8  divided in two groups of 4 members each..." and then talks about "a single group containing both A and B as a fraction of the total number of groups which can be formed"  
 All I can say is that the first line also just means "groups of 4 are made out of 8."

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