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## How many Handshakes???

tagged by: Brent@GMATPrepNow

This topic has 3 expert replies and 1 member reply
akhilsuhag Master | Next Rank: 500 Posts
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#### How many Handshakes???

Thu Aug 07, 2014 1:46 pm
There are 10 people in a room. If each person shakes hands with exactly 3 other people, what is the total number of handshakes?

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GMATGuruNY GMAT Instructor
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Thu Aug 07, 2014 2:17 pm
akhilsuhag wrote:
There are 10 people in a room. If each person shakes hands with exactly 3 other people, what is the total number of handshakes?

A) 15
B) 30
C) 45
D) 60
E) 120
The total number of pairs that can be formed from 10 people = 10C2 = 45.
If all of these pairs shake hands, each person in the room will shake hands with EVERY OTHER PERSON in the room.
The result will be 9 HANDSHAKES PER PERSON.
Since we want only 3 HANDSHAKES PER PERSON, we must divide by 3:
45/3 = 15.

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Brent@GMATPrepNow GMAT Instructor
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Thu Aug 07, 2014 2:57 pm
Quote:
There are 10 people in a room. If each person shakes hands with exactly 3 other people, what is the total number of handshakes?

A) 15
B) 30
C) 45
D) 60
E) 120
Here's an approach that doesn't require any counting techniques:

Each person shakes hands with exactly 3 other people
So, we have 10 people and each experiences 3 handshakes for a total of 30 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 30 by 2 to get 15

Here's a similar question: http://www.beatthegmat.com/count-questions-t274369.html

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Brent

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Rich.C@EMPOWERgmat.com Elite Legendary Member
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Thu Aug 07, 2014 9:21 pm
Hi akhilsuhag,

Since the answer choices are so "spread out", there's an interesting way to get to the correct answer by avoiding complex math and using "brute force" and a comparison:

We're told that there are 10 people in the room and that each person shakes hands with 3 other people.

Let's say there were 4 people, who we'll call A, B, C and D.

The handshakes would be:
AB
AC
BC
BD
CD

In this situation, each person shook hands with 3 people and there was a total of 6 handshakes.

If we TRIPLED the number of people, then we'd have 12 people, and we'd have TRIPLE the handshakes: 6 x 3 = 18.

Since we have FEWER than 12 people, we'll have FEWER than 18 handshakes. There's only one answer that fits:

GMAT assassins aren't born, they're made,
Rich

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madhusudhan237 Newbie | Next Rank: 10 Posts
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Wed Dec 23, 2015 6:02 am
HI Brent

So... if i want to generalize this hand shake problems,

in this case, we had 10 people & hand shakes with 3 other people .. so, its 10 x 3 = 60 / 2 ==> 30

assuming a problem asks for 10 people & hand shakes with 5 other people .. is it 10 x 5 = 50/2 = 25???

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