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sana.noor: Legendary Member; Posts: 512; Joined: Mon Jun 18, 2012 11:31 pmFollowed by: 20 members; Thanked: 42 times

by sana.noor » Mon Aug 05, 2013 10:49 am

There are 6 groups in a room. Each group consists of 3 men. How many handshakes will there be if each man only shakes hands with people who are outside his group?

i dont have the answer

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Source: Beat The GMAT — Problem Solving | Mon Aug 05, 2013 10:49 am

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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 » Mon Aug 05, 2013 11:12 am

sana.noor wrote:There are 6 groups in a room. Each group consists of 3 men. How many handshakes will there be if each man only shakes hands with people who are outside his group?

There are 18 men altogether.
So, if I'm in one of the 3-man groups, and I shake hands with people who are outside my group, I will shake hands with 15 people.
In fact, every man will shake hands with 15 people.
There are 18 men, so altogether there are (18)(15) handshakes. That's 270 handshakes.

BUT, there's a small problem with 270. Every handshake has been counted twice.
For example, person A shakes hands with 15 people, and one of these handshakes was with person B. At the same time, person B shakes hands with 15 people, and one of these handshakes was with person A. So, that handshake was counted twice.

Since every handshake is counted twice, we'll divide 270 by 2 to get a total of 135 handshakes

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Brent

Last edited by Brent@GMATPrepNow on Mon Aug 05, 2013 2:28 pm, edited 1 time in total.

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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 » Mon Aug 05, 2013 12:11 pm

Each handshake = one PAIR of men.
No handshake -- and thus no PAIR -- may be composed of two men from the SAME GROUP.
Thus, the question above can be rephrased as follows:

There are 6 groups in a room. Each group consists of 3 men. How many PAIRS of men can be formed if no pair may consist of two men from the same group?

Good pairs = total possible pairs - bad pairs.

Total possible pairs:
Number of pairs that can be formed from 18 men = 18C2 = (18*17)/(2*1) = 153.

Bad pairs:
A bad pair consists of two men from the SAME GROUP.
Number of pairs that be formed from each group of 3 men = 3C2 = (3*2)/(2*1) = 3.
Number of groups = 6.
To combine these options, we multiply:
3*6 = 18.

Thus:
Good pairs = 153-18 = 135.

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by [email protected] » Mon Aug 05, 2013 12:18 pm

Hi sana.noor,

Both Brent and Mitch have provided solid math explanations, provided you understand how the math "works."

Sometimes you have to just "brute force" the question though, and brute force can work pretty fast if you just get to work.

The 6 groups:
ABC
DEF
GHI
JKL
MNO
PQR

Since each man shakes hands with everyone outside of his group ONCE, you can keep a running tab of the total handshakes:
ABC - each man makes 15 handshakes = 45 handshakes
DEF - shook w/ABC already, so everyone else is left = 12 handshakes each = 36 handshakes
GHI - shook w/ABC and DEF, so everyone else is left = 9 handshakes each = 27 handshakes
JKL - shoot w/ABC, DEF and GHI, so everyone else = 6 handshakes each = 18 handshakes
MNO just has one group left = 3 handshakes each = 9 handshakes
PQR already shook everyone's hands

Total handshakes = 45 + 36 + 27 + 18 + 9 = 135 handshakes

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faraz_jeddah: Master | Next Rank: 500 Posts; Posts: 358; Joined: Thu Apr 18, 2013 9:46 amFollowed by: 7 members; Location: Jeddah, Saudi Arabia; Thanked: 42 times; GMAT Score:730

by faraz_jeddah » Tue Aug 06, 2013 6:05 am

[email protected] wrote:Hi sana.noor,

Both Brent and Mitch have provided solid math explanations, provided you understand how the math "works."

Sometimes you have to just "brute force" the question though, and brute force can work pretty fast if you just get to work.

The 6 groups:
ABC
DEF
GHI
JKL
MNO
PQR

Since each man shakes hands with everyone outside of his group ONCE, you can keep a running tab of the total handshakes:
ABC - each man makes 15 handshakes = 45 handshakes
DEF - shook w/ABC already, so everyone else is left = 12 handshakes each = 36 handshakes
GHI - shook w/ABC and DEF, so everyone else is left = 9 handshakes each = 27 handshakes
JKL - shoot w/ABC, DEF and GHI, so everyone else = 6 handshakes each = 18 handshakes
MNO just has one group left = 3 handshakes each = 9 handshakes
PQR already shook everyone's hands

Total handshakes = 45 + 36 + 27 + 18 + 9 = 135 handshakes

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

Great approach if you're not comfortable wit combination formulas.
Thumbs up!

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