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

GMATGuruNY wrote:

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.

/quote]

Hi Mick

How do 45 pairs contribute to 9 handshakes/person.
Please elaborate.

Thanks

[email protected] wrote:

GMATGuruNY wrote:

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.

/quote]

Hi Mick

How do 45 pairs contribute to 9 handshakes/person.
Please elaborate.

Thanks

Here's another way to look at it.

There are ten people in the room shaking hands, but each person does not shake hands with himself or herself. The person shakes hands with the other nine people in the room.

So from each of the ten people's perspectives, that person shakes hands with nine other people.

So in a way it seems as if this is what went on. 10 people x 9 handshakes = 90 handshakes total

But wait. When one person shakes hands, another person has also shaken hands. So, one handshake takes care of one handshake for each of two people. So there are not really 90 handshakes. There are only 45.

[email protected] wrote:Hi ahmedbari.ace,

Here is how you can 'visualize' 10 people each shaking hands with one another.

Let's name the people A,B,C,D,E F,G,H,I,J

The 45 handshakes would be....

AB,AC,AD,AE,AF,AG,AH,AI,AJ
BC,BD,BE,BF,BG,BH,BI,BJ
CD,CE,CF,CG,CH,CI,CJ
DE,DF,DG,DH,DI,DJ
EF,EG,EH,EI,EJ
FG,FH,FI,FJ
GH,GI,GJ
HI,HJ
IJ

You'll notice that there are 9+8+7+6+5+4+3+2+1 = 45 handshakes and that each letter appears in exactly 9 different handshakes.

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

[email protected] wrote:

GMATGuruNY wrote:
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.

Hi Mick

How do 45 pairs contribute to 9 handshakes/person.
Please elaborate.

Thanks

GMATGuruNY wrote:

[email protected] wrote:

GMATGuruNY wrote:
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.

Hi Mick

How do 45 pairs contribute to 9 handshakes/person.
Please elaborate.

Thanks

Let the 10 people be A, B, C, D, E, F, G, H, I, and J.
From these 10 people, a total of 45 pairs can be formed, as shown in my post above.

Person A is included in the following pairs:
AB, AC, AD, AE, AF, AG, AH, AI and AJ.
Thus, if all 45 pairs shake hands, A will shake hands with 9 other people.

This reasoning will hold true for B, C, D, E, F, G, H, I and J.
If all 45 pairs shake hands, B will shake hands with 9 other people.
If all 45 pairs shake hands, C will shake hands with 9 other people.
And so on.

Implication:
If all 45 pairs shake hands, each person will shake hands with 9 other people.
Dividing by 3, we get:
If 15 pairs shake hands, each person will shake hands with 3 other people.