Eight people are to be divided into two groups each having more than

Question

Eight people are to be divided into two groups each having more than or equal to 1 member. What is the probability that there will be 4 in each groups?

A. 1/7 
B. 35/127
C. 35/81
D. 5/11
E. 6/11

Regor60 · Accepted Answer

Nothing in the question stem suggests that the groups are distinguished by anything other than number, so the 8 can be split into:

1,7
2,6
3,5 and
4,4

One person can be selected:

8 ways. The other 7 become selected automatically.

Two people can be selected:

8C2 = 28 ways, the other 6 become selected automatically.

Three people can be selected:

8C5 = 56 ways, the other 5 become selected automatically.

Four people can be selected:

8C4 = 70 ways. However, because the two groups of 4 are being distinguished by their respective numbers where no distinction exists e.g.:

abcd efgh is the same as efgh abcd

The above 70 should be divided by 2:

35

So the total number of groups is:

8+28+56+35 = 127

And the probability is:

35/127

The correct answer is not among the answer choices.

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DivyaBachu · Answer

8 people can be divided into 2 groups
the group consists of 1 person 8C1=8 ways
the group consists of 2 people 8C2=28
the group consists of 3 people 8C3=56
the group consists of 4 people 8C4=70

the group consisting of 5 people becomes same as 3 people hence only above 4 possible.

total number of ways dividing a group is 162

probability of 4 in each group = 70/162 =35/81

Ans: C

deemat · Answer

Group A and Group B:

A 1, B 7
A = 8C1, B = 8C7 = 2(8)

A 2, B 6
A = 8C2, B = 8C6 = 2(28)

A 3, B 5
A = 8C3, B= 8C5 = 2(56)

A 4, B4
A = 8C4, B= 8C4 = 2(70)

Total number of ways to arrange =

2(8) + 2(28) + 2(56) + 2(70) = 324

Number ways to arrange 8 into two groups of 4:
140

Simplify: 140/324 = 35/81

C

8Harshitsharma · Answer

I have a difference in opinion to DivyaBachu, however, it's mathematics and there can be only one unambiguous answer to a properly worded problem. I did it like this:

8  different  people in 2  different  groups can be formed in  2^8   ways, but this includes zero people in each group so we need to exlude these 2 combinations, so total possible outcomes for us =  2^8 - 2  = 254

Now, calculating total favorable outcomes is simple. We need 4 people in each group and this can be done in  ^8C_4  ways =  \frac{8*7*6*5}{4*3*2*1}  i.e. 70

Probability =  \frac{70}{254}  i.e.  \frac{35}{127}

Please correct me if I am wrong. I did not assume that the groups were identical and I think we should not do that as the question doesn't state it explicitly, right?

ferrarce30 · Answer

Hi  deemat  ! 
Why in A 4 , B 4 do you multiply by two?
In the other cases I get it, for example in A 1 , B 7, it could be that A is a group of 1 and B of 7 or that A is a group of 7 and B of 1.
But in A 4 , B 4 isn't there only 1 case?

siddhantvarma · Answer

To find the probability that eight people are divided into two groups with exactly four members in each, we can use the following steps:

Total ways to divide the group into two subgroups:

Since the two groups are indistinguishable, each unique grouping of 4 people can only be paired with the remaining 4 people in one way. The combination formula gives the total number of ways to select 4 people out of 8: 8C4

Since the order of the groups doesn't matter: 8C4/2 = 35

Total ways to divide into any two groups with at least 1 member in each:

The total number of ways to divide 8 people into two non-empty groups is 2^8−2, which accounts for all possible ways to assign each person to a group minus the two cases where all are in one group (either all 8 in one group and 0 in the other, which are not allowed):

However, since the groups are indistinguishable, each division is counted twice, so we need to divide by 2:

2^8−2=254 (subtracting the two cases where one group is empty)Since we count each partition twice (Group A and Group B are indistinguishable):

254/2=127

Probability calculation:

The probability P of dividing the 8 people into two groups of exactly 4 members each is the ratio of the number of favourable outcomes to the total number of possible divisions:

P=Number of favorable outcomes/Total number of possible divisions = 35/127

So, the correct answer is: 35/127 (B)

bumpbot · Answer

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