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A rectangular table seats 4 people on each of two sides, with every person directly facing another person across the table. If eight people choose their seats at random, what is probability that any two of them directly face other?
(A) 1/56
(B) 1/8
(C) 1/7
(D) 15/56
(E) 4/7
(A) 1/56
(B) 1/8
(C) 1/7
(D) 15/56
(E) 4/7
anaik100
Person A can take any place, probability that person B will take the opposite seat is 1/7 (as there are 7 seats left).
Answer: C.
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Posts: 3
We are given :
+ + + +
TABLE
+ + + +
Probability = # Favorable Outcomes / # Total Outcomes
Total Outcomes = 8 * 7 (Since a spot on the table for 2 people can be selected in 8 * 7 ways) = 56
Favorable Outcomes = 8 * 1 (One person can pick a seat in 8 ways, but to sit opposite to that person, there is only 1 option, thus 8*1) = 8
Probability = 8/56 = 1/7, thus the Answer is C.
Thanks for the Kudos!
+ + + +
TABLE
+ + + +
Probability = # Favorable Outcomes / # Total Outcomes
Total Outcomes = 8 * 7 (Since a spot on the table for 2 people can be selected in 8 * 7 ways) = 56
Favorable Outcomes = 8 * 1 (One person can pick a seat in 8 ways, but to sit opposite to that person, there is only 1 option, thus 8*1) = 8
Probability = 8/56 = 1/7, thus the Answer is C.
Thanks for the Kudos!
General Discussion
A B C D
------------
| TABLE |
------------
E F G H
A to H are 8 people.
Prob to select any 1 person = 1
Prob to select the person opposite to the chosen person = 1/7
For ex. If we select A as the person than prob of choosing E is 1/7.
Hence, answer will be C.
------------
| TABLE |
------------
E F G H
A to H are 8 people.
Prob to select any 1 person = 1
Prob to select the person opposite to the chosen person = 1/7
For ex. If we select A as the person than prob of choosing E is 1/7.
Hence, answer will be C.
Bunuel
The question asks for the probability of any two people (say A and B) facing each other.
Method 1:
A can take any seat. After that B should take the seat facing A which is 1 of the 7 remaining seats. So probability of A and B facing each other = 1/7
Method 2:
In how many ways can 8 people can sit on a rectangular table with 4 chairs on 2 opposite sides? The first person can sit in 4 ways (there are 4 distinct seats with respect to the table. Think circular arrangement). Now there are 7 distinct seats and 7 people so they can sit in 7! ways.
Total number of ways = 4*7!
In how many ways can 8 people sit such that A and B face each other? A goes and sits in 4 ways (as before). B sits directly opposite him in 1 way only. Now we have 6 distinct seats and 6 people. They can sit in 6! ways.
Probability = 4*6!/4*7! = 1/7
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anaik100
Consider those 2 persson A and B so -
Fav condition:
A can sit in any of seat around the table no of ways - 8
B has to take seat opposite to A no of ways - 1
other member can take seat other 6 seats no of ways - 6!
All condition:
all possible combination without restriction no of ways - 8!
\(prob = fav/all => 8*1*6!/8!\)
\(prob = 1/7\)
shashankism
Joined: 13 Mar 2017
Last visit: 29 Jun 2026
Posts: 612
Affiliations: IIT Dhanbad
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE:Engineering (Energy)
Posts: 612
VeritasPrepKarishma
To solve this question we can do it by 2 methods....
Let the 2 persons facing each other be A & B
Method 1 : Place A at a position... Now there are 7 spaces for B to seat but in only one case it is opposite to B. So probability of B facing A = 1/7.
Method 2 : Counting total no. of arrangements
So, Total no. of ways of seating all the 8 people = 8!
(VeritasPrepKarishma I disagree with you here for total no. of ways. Lets A is seated at corner seat then left and right seating of A is very different in nature. Hence it may not be considered as circular arrangement.)
Total no. of desired ways in which A & B faces each other = 4 (pair of opposite seats) * 2 (arranging A and B on the pair of opposite seat) * 6! (arranging other 6 people on the 6 seats) = 4*2*6!
So required probability = 4*2*6!/8! = 1/7
Answer C
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shashankism
Look at this:
Attachment:
Fig1.jpeg [ 22.95 KiB | Viewed 10102 times ]
Using same logic for the shorter sides of the rectangle too, we will get 4 unique places to sit for the first person, not 8 (with respect to the table)
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