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There are 10 people in a room. If each person shakes hands with exactl

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There are 10 people in a room. If each person shakes hands with exactly 3 other people, what is the total number of handshakes?

15
30
45
60
120

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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 25 Nov 2010, 21:33
shrive555 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?

15
30
45
60
120


Consider one hand that gets shaken as one event. One person shakes his hand with three other people so 3 events take place per person. In all 10*3 = 30 events take place. But two hands make one handshake. So 30 of these events will make 15 handshakes.

Question: Substitute 5 in place of 10. What do you get? Why?
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 26 Nov 2010, 10:13
why do we have to divide 30 by 2...
substituting 5 with 10, i would get 5 which is half of 10.
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 26 Nov 2010, 12:57
Hello Karishma

Can you please explain further?
I think I don't quite understand it.

Is is so, that when you want to shake hand with 3 people, the number of persons in the group has to be even and equal or greater than 4?

Thanks

VeritasPrepKarishma wrote:
shrive555 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?

15
30
45
60
120


Consider one hand that gets shaken as one event. One person shakes his hand with three other people so 3 events take place per person. In all 10*3 = 30 events take place. But two hands make one handshake. So 30 of these events will make 15 handshakes.

Question: Substitute 5 in place of 10. What do you get? Why?

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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 26 Nov 2010, 15:47
shrive555 wrote:
why do we have to divide 30 by 2...
substituting 5 with 10, i would get 5 which is half of 10.


Consider you and I shake hands. When I count 30, I have counted it twice. Once for you, and once for me. But it is actually just one handshake. Each one of those 30 events was one hand that was shaken. Two of those events make one handshake so we divide by 2.
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 26 Nov 2010, 16:07
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craky wrote:
Hello Karishma

Can you please explain further?
I think I don't quite understand it.

Is is so, that when you want to shake hand with 3 people, the number of persons in the group has to be even and equal or greater than 4?

Thanks

VeritasPrepKarishma wrote:
shrive555 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?

15
30
45
60
120


Consider one hand that gets shaken as one event. One person shakes his hand with three other people so 3 events take place per person. In all 10*3 = 30 events take place. But two hands make one handshake. So 30 of these events will make 15 handshakes.

Question: Substitute 5 in place of 10. What do you get? Why?


Ok Look.

4 people: shrive555, craky, karishma and Mr X
We have to shake hands in this group such that each person shakes hands with 3 people.

So shrive555 starts:
shrive555 - craky : 1 handshake but 2 hands were shaken. shrive555's and craky's
shrive555 - karishma : 1 handshake but 2 hands were shaken. shrive555's and karishma's
shrive555 - Mr X : 1 handshake but 2 hands were shaken. shrive555's and Mr X's

Now shrive555 has shaken hands with 3 people. There were 3 handshakes. But 6 hands were shaken.
Now, when all 4 of us shake hands with 3 people, each person's hand will be shaken 3 times. i.e. in all 12 hands will be shaken. But they will add up to only 6 handshakes.
The other 3 handshakes will be:
craky - karishma
craky - Mr X
karishma - Mr X
So all of us have shaken hands with exactly 3 people.

Similarly when there are 10 people, each person shakes his hand 3 times. So in all 30. But 2 of these hands combined to make one handshake. So we will get only 15 total handshakes.

Now when there are 5 people and each person has to shake hands with exactly 3 people, each person will shake his hand 3 times. We will have total 15 hands that will be shaken. So how many handshakes does it make? 7.5? That is not possible. This is because it is not possible for 5 people to shake hands such that each person will shake hands with exactly 3 people. Can you think of the condition which must be satisfied such that it is possible that each person shakes hands with exactly 3 people? (craky, you are there. Just think some more to be clear.)
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 19 Mar 2016, 15:08
VeritasPrepKarishma wrote:
shrive555 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?

15
30
45
60
120


Consider one hand that gets shaken as one event. One person shakes his hand with three other people so 3 events take place per person. In all 10*3 = 30 events take place. But two hands make one handshake. So 30 of these events will make 15 handshakes.

Question: Substitute 5 in place of 10. What do you get? Why?
[/quote]

Ok Look.

4 people: shrive555, craky, karishma and Mr X
We have to shake hands in this group such that each person shakes hands with 3 people.

So shrive555 starts:
shrive555 - craky : 1 handshake but 2 hands were shaken. shrive555's and craky's
shrive555 - karishma : 1 handshake but 2 hands were shaken. shrive555's and karishma's
shrive555 - Mr X : 1 handshake but 2 hands were shaken. shrive555's and Mr X's

Now shrive555 has shaken hands with 3 people. There were 3 handshakes. But 6 hands were shaken.
Now, when all 4 of us shake hands with 3 people, each person's hand will be shaken 3 times. i.e. in all 12 hands will be shaken. But they will add up to only 6 handshakes.
The other 3 handshakes will be:
craky - karishma
craky - Mr X
karishma - Mr X
So all of us have shaken hands with exactly 3 people.

Similarly when there are 10 people, each person shakes his hand 3 times. So in all 30. But 2 of these hands combined to make one handshake. So we will get only 15 total handshakes.

Now when there are 5 people and each person has to shake hands with exactly 3 people, each person will shake his hand 3 times. We will have total 15 hands that will be shaken. So how many handshakes does it make? 7.5? That is not possible. This is because it is not possible for 5 people to shake hands such that each person will shake hands with exactly 3 people. Can you think of the condition which must be satisfied such that it is possible that each person shakes hands with exactly 3 people? (craky, you are there. Just think some more to be clear.)[/quote]


Hi Karishma,

Take this forward for 10 individuals we have the following scenario:

1 & 2, - 1st unique handshake
1 & 3, - 2nd unique handshake
1 & 4, - 3rd unique handshake
2 & 1,
2 & 3, - 4th unique handshake
2 & 4, - 5th unique handshake
3 & 1,
3 & 2,
3 & 4, - 6th unique handshake

Case for individual number 5, 6, 7 and 8 will also make 6 unique handshakes. So in all 12 upto now.

For individual number 9 and 10.

9 & 10 - 11th unique handshake
10 & 9

Now, individuals numbered 9 and 10 don't even end up meeting the condition of "exactly 3 handshakes".

I selected 15, as it was the closest (to 11 unique handshakes) choice. Can you please assist? Thanks.
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 19 Mar 2016, 20:36
Meetup wrote:

Hi Karishma,

Take this forward for 10 individuals we have the following scenario:

1 & 2, - 1st unique handshake
1 & 3, - 2nd unique handshake
1 & 4, - 3rd unique handshake
2 & 1,
2 & 3, - 4th unique handshake
2 & 4, - 5th unique handshake
3 & 1,
3 & 2,
3 & 4, - 6th unique handshake

Case for individual number 5, 6, 7 and 8 will also make 6 unique handshakes. So in all 12 upto now.

For individual number 9 and 10.

9 & 10 - 11th unique handshake
10 & 9

Now, individuals numbered 9 and 10 don't even end up meeting the condition of "exactly 3 handshakes".

I selected 15, as it was the closest (to 11 unique handshakes) choice. Can you please assist? Thanks.


Since it is given that each person does shake hands with exactly 3 people, it will certainly be possible. Here is one such instance:

1&2
3&4
5&6
7&8
9&10
(Each person shakes hands once)

1&3
2&5
4&7
6&9
8&10
(Each person shakes hand with another person)

1&5
2&9
3&7
4&10
6&8
(Each person shakes hands with yet another person)
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 24 Apr 2017, 05:12
Hello Karishma!

Quote:
Question: Substitute 5 in place of 10. What do you get? Why?


9.
Let's say we have 5 people in the room: A, B, C, D, E.
And there are handshakes:
A: B, C, D
B: C, D, E
C: D, E
D: E
So, there are will be 3+3+2+1=9 handshakes in total. Is that correct?
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 12 Jun 2017, 11:58
VeritasPrepKarishma wrote:
shrive555 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?

15
30
45
60
120


Consider one hand that gets shaken as one event. One person shakes his hand with three other people so 3 events take place per person. In all 10*3 = 30 events take place. But two hands make one handshake. So 30 of these events will make 15 handshakes.

Question: Substitute 5 in place of 10. What do you get? Why?

Can you please suggest more questions like this one to practice?
I struggled with this one really..my approach was different for solving this. Your approach is really creative, and without having solved something of this sort before or read about the method you used, I wouldn't have been able to do this question. Please also suggest something to read.
Thank you so much :)
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 12 Jun 2017, 13:36
ShashankDave wrote:
VeritasPrepKarishma wrote:
shrive555 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?

15
30
45
60
120


Consider one hand that gets shaken as one event. One person shakes his hand with three other people so 3 events take place per person. In all 10*3 = 30 events take place. But two hands make one handshake. So 30 of these events will make 15 handshakes.

Question: Substitute 5 in place of 10. What do you get? Why?

Can you please suggest more questions like this one to practice?
I struggled with this one really..my approach was different for solving this. Your approach is really creative, and without having solved something of this sort before or read about the method you used, I wouldn't have been able to do this question. Please also suggest something to read.
Thank you so much :)


http://gmatclub.com/forum/how-many-diag ... 01540.html
http://gmatclub.com/forum/if-10-persons ... 10622.html
http://gmatclub.com/forum/10-business-e ... 26163.html
http://gmatclub.com/forum/how-many-diff ... 29992.html
http://gmatclub.com/forum/15-chess-play ... 55939.html
http://gmatclub.com/forum/there-are-5-c ... 27235.html
http://gmatclub.com/forum/if-each-parti ... 42222.html
http://gmatclub.com/forum/there-are-8-t ... 34582.html
http://gmatclub.com/forum/there-are-8-t ... 32366.html
http://gmatclub.com/forum/in-a-kickball ... 61846.html
https://gmatclub.com/forum/if-11-person ... 89706.html
https://gmatclub.com/forum/6-people-mee ... 18398.html
https://gmatclub.com/forum/there-are-10 ... 05386.html
https://gmatclub.com/forum/there-are-in ... 07442.html
https://gmatclub.com/forum/during-a-sea ... 59677.html
https://gmatclub.com/forum/six-three-re ... 31082.html

Hope it helps.
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There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 12 Jun 2017, 18:55
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shrive555 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?

15
30
45
60
120


if each of the 10 had shaken hands once with 9 others,
there would have been 10*9/2, or 45 total handshakes,
but, because only one third (3 out of 9) of possible handshakes take place,
there are only one third (15 out of 45) total handshakes
15
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 13 Jun 2017, 00:12
craky wrote:
Hello Karishma

Can you please explain further?
I think I don't quite understand it.

Is is so, that when you want to shake hand with 3 people, the number of persons in the group has to be even and equal or greater than 4?

Thanks

VeritasPrepKarishma wrote:
shrive555 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?

15
30
45
60
120


Consider one hand that gets shaken as one event. One person shakes his hand with three other people so 3 events take place per person. In all 10*3 = 30 events take place. But two hands make one handshake. So 30 of these events will make 15 handshakes.

Question: Substitute 5 in place of 10. What do you get? Why?


If all 5 people shook hands with the other 4 - a total of 10 handshakes would have taken place (5*4/2).
now for each to shake exactly three hands- there would be a total of 9 or more handshakes- as either one would shake with only two or one would shake with four. please explain
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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 11 Feb 2018, 13:45
Hi All,

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:

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Re: There are 10 people in a room. If each person shakes hands with exactl  [#permalink]

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New post 22 Apr 2018, 06:35
Top Contributor
shrive555 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


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

Cheers,
Brent
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