How many Handshakes???

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akhilsuhag: Master | Next Rank: 500 Posts; Posts: 351; Joined: Mon Jul 04, 2011 10:25 pmFollowed by: 4 members; Thanked: 57 times

by akhilsuhag » 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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Source: Beat The GMAT — Problem Solving | Thu Aug 07, 2014 1:46 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » 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.

The correct answer is A.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Aug 07, 2014 2:57 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?

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

Answer: A

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

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by [email protected] » 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
AD
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:

Final Answer: A

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madhusudhan237: Newbie | Next Rank: 10 Posts; Posts: 5; Joined: Fri Oct 30, 2015 12:35 pm

by madhusudhan237 » 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???

Thanks for your feedback

Madhu

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