In a room are five chairs to accommodate 3 people. One person to a cha

Question

In a room are five chairs to accommodate 3 people. One person to a chair. How many seating arrangements are possible?

(A) 45
(B) 60
(C) 72
(D) 90
(E) 120

KarishmaB · Accepted Answer

Think of it this way:

The first person has 5 options i.e. 5 chairs to choose from
The second person has 4 options i.e. 4 chairs to choose from (first one has already taken a chair)
The third person has 3 options i.e. 3 chairs to choose from (two chairs are already gone)

Total seating arrangements 5*4*3 = 60

Or
Assume that the two vacant chairs will be occupied by 2 people,both named V.
Now you have to arrange 5 people on 5 chairs but two of them are identical (they are both V - vacant)
You can do this in 5!/2! = 60 ways

Bunuel · Answer

Another method: C^3_5*3!=60 , where C^3_5 is the number of ways to choose which 3 chairs will be occupied out of 5 and 3! is the number of ways to arrange 3 people. Answer: B. P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to the rules 3 and 7. Thank you.

Bunuel · Answer

C^3_5*3!=60 , where  C^3_5  is the number of ways to choose 3 chairs for 3 people, and 3! is the number of arrangements of 3 people on those chairs.

Or directly:  P^3_5=60 : choosing 3 out of 5, where order matters.

Answer: B.

holdem · Answer

I actually solved this problem by the following thoughts (derived from factorials):
First Person: 5 Chairs to choose
Second Person: 4 Chairs to choose
Third Person: 3 Chairs to choose
Then I simply multiplied this values and got the answer. Basically it’s 5! – 2!.

Now the interesting part: I just did that because I had no other idea, how to tackle this problem. Would someone please tell me, if I was just lucky? (and probably explain it differently)
Unfortunately I can’t understand what you are exactly calculating in your solution Bunuel.

KarishmaB · Answer

You used the Basic Counting Principle which is absolutely fine. 
Answer is 5*4*3 = 60 (which by the way, is not the same as 5! - 2! = 118)

GMATMBA5 · Answer

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Why the two empty chairs are not accounted for? I mean why it's not  C^3_5*3!*2!  , where 2! is the empty chairs. Because the three person can have 2 adjacent empty chairs in between them or no empty chair or alternate empty chairs.

Bunuel · Answer

I don;t understand what you mean. The chairs are standing in a row. We are accommodating 3 people.

GMATMBA5 · Answer

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Why the two empty chairs are not accounted for? I mean why it's not  C^3_5*3!*2!  , where 2! is the empty chairs. Because the three person can have 2 adjacent empty chairs in between them or no empty chair or alternate empty chairs.

I don;t understand what you mean. The chairs are standing in a row. We are accommodating 3 people.

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Suppose P1C, P2C and P3C denote three persons sitting on three different chairs and EC denotes an empty chair.

With  C^3_5*3! , we determined the pattern of three persons sitting on 5 chairs, with interchanging seats.
But is the seating arrangement P1C-EC-P2C-EC-P3C not different from P1C-P2C-EC-EC-P3C. Do we not have to consider empty chairs as well for forming the pattern or is it that which chairs are left empty doesn't matter.

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chetan2u · Answer

Hi...

when you are choosing 3 chairs out of 5 by 5C3, you are already catering for where the E are ..
so FFFEE is taken as a different scenario than FFEEF....
Now these three can be filled in 3! ways that is why 5C3*3!

let three person be A, B and C
here 3!=3*2=6 means FFFEE is
ABCEE
ACBEE
BACEE
BCAEE
CABEE
CBAEE

hope it helps

ManifestDreamMBA · Answer

Person 1 can be assigned to any of the 5 chairs
Person 2 can be assigned to remaining 4 chairs only
Person 3 can be assigned to remaining 3 chairs only

Total options = 5*4*3 = 60

In a room are five chairs to accommodate 3 people. One person to a cha

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In a room are five chairs to accommodate 3 people. One person to a cha

Prep Toolkit

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