In a conference of 3 delegates from each of 8 different organizations,

Question

In a conference of 3 delegates from each of 8 different organizations, each delegate shook hands with every person other than those from his or her own organization. How many handshakes took place in the conference?

(A) 48
(B) 96
(C) 252
(D) 270
(E) 504

(A) 48
(B) 96
(C) 252
(D) 270
(E) 504

ScottTargetTestPrep · Accepted Answer

We are given that there are 3 representatives from 8 different companies. So, there are a total of 24 representatives.

If every representative were to shake hands with all other representatives (meaning all 24 reps would shake hands), this would happen in the following number of ways:

24C2 = (24 x 23)/2! = 12 x 23 = 276 ways

However, since each person shook hands with every person not from his or her own company, 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 8 companies, this would occur 8 x 3 = 24 times.

Thus, the number of ways for the reps to shake hands with every person not from his or her own company is 276 - 24 = 252 ways.

Alternate Solution:

Each of the 3 x 8 = 24 representative will shake hands with 24 - 3 = 21 people since representatives within the same organization won’t shake hands with each other. Thus, 24 x 21 will be equal to twice the number of handshakes that took place in the organization, since each handshake is counted twice (each handshake is counted once for each person involved in the handshake, hence the double count). Then, there were 24 x 21 x (1/2) = 12 x 21 = 252 handshakes that took place in the organization.

Answer: C

Salsanousi · Answer

This is a tricky one.

1,1,1
2,2,2
3,3,3 etc..

If the delegates from group (1) shake hands with each from group (2) this means 3 hand shakes for each of the delegates from Group 1.

Since delegates from group (1) will shake hands with all delegates from 2,3,4,5,6,7,8

It means each member in group 1 will shake hands 21 times. (8-1)*3 = 21

8-1 = 7 to exclude the hand shakes from their group.

Since each shakes hand once it means per group they have 21*3 = 63

There are 8 groups so it means 63*8 = 504
But this double-counts.

So we divide it by 2 we get 504/2 = 252.

Answer choice C.

Another way would be to say

A member of group one gets 21 hand shakes
A member of group two gets 18
A member of group three gets 15
A member of group four gets 12
A member of group five gets 9
A member of group six gets 6
A member of group seven gets 3
And a member of group 8 gets 0.

Adding all the above numbers we get 84

And 84*3 = 252

Here the 3 is to account for 3 members in the group.

I hope it is clear

Posted from my mobile device

ArjunJag1328 · Answer

1 company = 3 delegates
Total companies present = 8
Total number of delegates = 24

In the question, each delegate shook hands with every person other than those from his or her own organization
=>1 delegate will shake hands with 21 other delegates
=>Total =  21 handshakes

There are a total of 24 delegates and therefore the total number of handshakes = 21x24 =  504 Handshakes (This includes the repetition of handshakes)
So we should divide it by 2
=>504/2
=>252

Hence C is the answer

KarishmaB · Answer

Consider the 3 delegates of the first org. Each of the 3 delegates will shake hands with delegates from the other 7 organisations i.e. with 7*3 people.
Consider the next team of delegates from the second org. Each of the 3 delegates will shake hands with delegates of other 6 orgs (since they have already shaken hands with the first org delegates) i.e. with 6*3 = 18 people. 
So we see the pattern: No of handshakes = 3*( 21 + 18 + ... + 3 )

The underlined is an AP. Its sum = (No of terms)*(First term + last term)/2

Sum = 3 * 7*(21 + 3)/2 = 3*84 = 252

Answer (C)

N0BU · Answer

KarishmaB   Bunuel  is the below approach correct?

KarishmaB · Answer

That's correct too.

N0BU · Answer

Thankyou  KarishmaB  !!

riri3026 · Answer

If we had to count the handshakes with the delegation that the person belonged to, then we would have 8 * 3 = 24 ( for the number of handshakes)
then, we would take the number of people shaking their hands = 8 *3 again, as 24 people

so 24 * 24 / 2 (as to exclude the double counting) = 288

Can anyone confirm this? (note: I have excluded the part of each delegate shook hands with every person other than those from his or her own organization, just to try this way out)

Bunuel · Answer

Total - Restriction approach would be:
 
24C2 - 8*3 = 276 - 24 = 252.
 
Answer: C.

pierjoejoe · Answer

hi!
i'd like someone to take a look to my answer.

there are 8 groups of 3, meaning that each group will interact with another group in 8C2 = 28
we have 28 ways to create couples of groups from 8 groups. each couple of groups will "generate" C(6,2) - 2*C(3,2) = 9 (or put more simply 3^2) handshakes.
in total we have 28 * 9 handshakes = 252

Bunuel · Answer

_________________
That's correct.

In a conference of 3 delegates from each of 8 different organizations,

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