Six three-representative delegations attend an international conferenc

Question

Six three-representative delegations attend an international conference. The representatives shake hands when they are introduced to one another. How many handshakes are possible if each delegate shakes hands only once with every other attendant except with those of his/her delegation?

A) 36
B) 72
C) 90
D) 135
E) 388

A) 36 
B) 72 
C) 90 
D) 135 
E) 388

vitaliyGMAT · Accepted Answer

Total number of handshakes:  _{18}C_2 = 153

We need to subtract from this the number of handshakes between the members of each delegation

_3C_2 = 3

and because we have 6 delegations we need o subtract  3*6 = 18  cases

Our final answer: 153 – 18 = 135

Answer D

RMD007 · Answer

IMO D.

There are 6 delegation, each contains 3 representatives.

Consider 1st delegation:
Member 1 from delegation 1 will handshake with every other person in other 5 delegations. So total handshakes = no of people in other delegations except 1st = 15.
Similarly logic for other two members of 1st delegation.
So, total handshakes from delegation 1 = 15*3 = 45.

Delegation 2:
For members in delegation 2, consider delegation members other than 1 & 2 . Remaining members - 12 (4 delegations) 
Total handshakes - 12*3 = 36.

So applying same logic
Total handshakes = 15*3 + 12*3 + 9*3 + 6*3 + 3*3 = 45+36+27+18+9 = 135.

Tip: No need to consider handshakes of last delegation. Since the same has been already covered in previous cases.

AR15J · Answer

Thanks Ruchi for writing this "There are 6 delegations, each contains 3 representatives." Now I can comprehend the ques.
(5*3*3*6)/2=135

shashank1tripathi · Answer

I think answer should be D.
Total number of members: 6 * 3 = 18
Total number of handshakes(including among same delegations) : 18C2 = 153
Now, Total handshakes among same delegations = 6 * 3C2 = 18
So, Answer = 153 - 18 = 135

US09 · Answer

There is another way to solve this: each member shakes hand with other members and not with herself, therefore each member shakes hands with 15 members. So, 18 members shake hands with 15 others= 18*15. But when one member shakes hand with other, the other will not be required to repeat the same (since, shaking hands is a two way thing, does not need to be repeated once done between two members. Hence, 18*15/2= 135.

ScottTargetTestPrep · Answer

We are given that there are 3 representatives from 6 different delegations. So, there are a total of 18 representatives.

If every representative were to shake hands with all other representatives (meaning all 18 reps would shake hands), this would happen in the following number of ways:

18C2 = (18 x 17)/2! = = 9 x 17 = 153 ways

However, since each person shook hands with every person not from his or her own delegation, we can subtract out the number of times those handshakes occurred.

Since each company has 3 reps, the number ways those three reps can shake hand is 3C2 = (3 x 2)/2! = 3 ways, and since there are 6 companies, this would occur 6 x 3 = 18 times.

Thus, the number of ways for the reps to shake hands with every person not from his or her own delegation is 153 - 18 = 135 ways.

Alternate Solution:

Let’s count the total number of handshakes. Since each of the 6 x 3 = 18 representatives shake hands with 5 x 3 = 15 other representatives, this makes 18 x 15 = 270 handshakes. However, we counted each handshake twice, once for each of the parties involved in the handshake. Therefore, the number of handshakes that take place is 270 / 2 = 135.

Answer: D

Archit3110 · Answer

total handshakes would 18c3 ; 153
total hand shakes if done within each group ; 3c2 *6 ; 18
so total handshakes shakes hands only once with every other attendant except with those of his/her delegation ; 153-18 ; 135
IMO D

TestPrepUnlimited · Answer

If every single person took turns shaking hands with all other people, we would have 18 * 15 handshakes in total (15 because "each delegate shakes hands only once with every other attendant except for those of his/her delegation"). However, with this product we would have any two people (of different delegation) shaking hands with each other twice, e.g. A shakes hands with B and B shakes hands with A are both counted with this method. We only need one of each combination to happen therefore we divide by 2 to get 9*15 = 135.

Ans: D

Kinshook · Answer

Given: Six three-representative delegations attend an international conference. The representatives shake hands when they are introduced to one another. 
Asked: How many handshakes are possible if each delegate shakes hands only once with every other attendant except with those of his/her delegation?

Every delegate will shake hands with = 5*3 = 15 delegates
Since there are 18 delegates, total handshakes = 18*15 = 135*2

Since each handshake is counted twice, total handshakes = 135

IMO D

Aravind04 · Answer

1. Total Handshakes possible = N(N-1)/2 ==> (18*17)/2 ==> 153

2. In one group there can be 2 hand shakes from one person to other 2 people say delegation one has ABC as members
a handshakes with AB,AC so other 2 also handshake with others with in delegation

3. total 6 hand shakes with in delegation divided by 2 will eliminate counting twice 
hence==> 3 hand shakes per delegation * 6 delegations = 18 hand shakes which should not be counted

4. Total -handshakes with in delegations ==>153-18==> 135

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Six three-representative delegations attend an international conferenc

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Six three-representative delegations attend an international conferenc

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards