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Question

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

Brent@GMATPrepNow · Answer

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

Cheers,
Brent

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

Brent Hanneson - Creator of GMATPrepNow.com

GMATGuruNY · Answer

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.

Rich.C@EMPOWERgmat.com · Answer

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

Contact Rich at [email protected]

faraz_jeddah · Answer

Rich.C@EMPOWERgmat.com 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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