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# A group of 12 people who have never met are in a classroom.

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Joined: 07 Jun 2014
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A group of 12 people who have never met are in a classroom. [#permalink]  30 Jul 2018, 10:52
Expert's post
00:00

Question Stats:

76% (00:34) correct 24% (01:28) wrong based on 25 sessions
A group of 12 people who have never met are in a classroom. How many handshakes are exchanged if each person shakes hands exactly once with each of the other people in the room?

(A) 12
(B) 22
(C) 66
(D) 132
(E) 244
[Reveal] Spoiler: OA

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Sandy
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Re: A group of 12 people who have never met are in a classroom. [#permalink]  31 Jul 2018, 08:28
how to solve this?
GRE Prep Club Legend
Joined: 07 Jun 2014
Posts: 4810
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 117

Kudos [?]: 1895 [0], given: 397

Re: A group of 12 people who have never met are in a classroom. [#permalink]  08 Aug 2018, 18:09
Expert's post
Explanation

Multiple approaches are possible here. One way is to imagine the scenario and count up the number of handshakes. How many hands does everyone need to shake?

There are 11 other people in the room, so the first person needs to shake hands 11 times. Now, move to the second person: how many hands must he shake? He has already shaken one hand, leaving him 10 others with whom to shake hands.

The third person will need to shake hands with 9 others, and so on. Therefore, there are a total of 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 handshakes. The fastest way to find the sum of a group of consecutive numbers is to take the average of the first and last terms and multiply it by the number of terms.

The average is $$\frac{11+1}{2}= 6$$ and there are $$11 - 1 + 1 = 11$$ terms (find the difference between the terms and “add one before you’re done”). The sum is $$6 \times 11 = 66$$.
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Sandy
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Re: A group of 12 people who have never met are in a classroom. [#permalink]  09 Aug 2018, 13:01
1
KUDOS
Expert's post
sandy wrote:
A group of 12 people who have never met are in a classroom. How many handshakes are exchanged if each person shakes hands exactly once with each of the other people in the room?

(A) 12
(B) 22
(C) 66
(D) 132
(E) 244

Here's an approach that doesn't require any counting techniques:

person shakes hands exactly once with each of the other people in the room
So, each person shakes hands with 11 people.
So, we have 12 people, and each experiences 11 handshakes, for a total of 132 handshakes.

IMPORTANT: at this point, we need to recognize that every handshake has been counted TWICE.
For example, if Person A and Person B shake hands, then Person A counts it as a handshake, AND Person B also counts it as a handshake.
Of course only one handshake occurred.

To account for the DUPLICATION, we'll divide 132 by 2 to get 66 handshakes

Cheers,
Brent
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Re: A group of 12 people who have never met are in a classroom. [#permalink]  09 Aug 2018, 13:06
2
KUDOS
Expert's post
sandy wrote:
A group of 12 people who have never met are in a classroom. How many handshakes are exchanged if each person shakes hands exactly once with each of the other people in the room?

(A) 12
(B) 22
(C) 66
(D) 132
(E) 244

Another approach is to ask "In how many different ways can we select 2 people from 12 people?"
The idea here is that, for each unique selection of 2 people, we can get those people to shake hands.

The order in which we select the two people does not matter. For example, selecting Person A 1st and Person B 2nd is exactly the same as selecting Person B 1st and Person A 2nd.
Since the order in which we select the two people does not matter, we can use combinations.

We can select 2 people from 12 people in 12C2 ways.
12C2 = 66

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Re: A group of 12 people who have never met are in a classroom.   [#permalink] 09 Aug 2018, 13:06
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