M07-27

Question

At a conference, there are 10 business executives and 7 chairmen. If each business executive shakes hands with every other executive and each chairman exactly once, and each chairman shakes hands only with the business executives (but not with other chairmen), how many handshakes take place?

A. 144
B. 131
C. 115
D. 90
E. 45

Show Answer

Official Answer

C

A. 144
B. 131
C. 115
D. 90
E. 45

Bunuel · Accepted Answer

Official Solution:

At a conference, there are 10 business executives and 7 chairmen. If each business executive shakes hands with every other executive and each chairman exactly once, and each chairman shakes hands only with the business executives (but not with other chairmen), how many handshakes take place?

A. 144
B. 131
C. 115
D. 90
E. 45

Show Answer

Official Answer

C

A. 144
B. 131
C. 115
D. 90
E. 45

Approach 1: 
 
The total number of possible handshakes between 10 business executives and 7 chairmen (without any restrictions) is equivalent to the number of unique pairs we can form from these 17 individuals, which is  C^2_{17} . Similarly, the number of handshakes just between the chairmen is  C^2_{7}  (restricted handshakes).
 
Thus, the desired total is:  	ext{Total handshakes} - 	ext{Restricted handshakes} = C^2_{17} - C^2_{7} = 136 - 21 = 115 .
 
 Approach 2: 
 
Another method involves directly calculating the number of handshakes between the executives, which is  C^2_{10} , and then adding the handshakes between the executives and the chairmen, which is  10 * 7  (as each of the 10 executives shakes the hand of each of the 7 chairmen). So, the total is:  C^2_{10} + 10 * 7 = 115 .

Answer: C

US09 · Answer

Another way to approach this is: When a group shakes hands within itself, then the number of handshakes are counted twice as when A shakes hands with B, B is also shaking hands with A. Hence, easy way is to divide the total number of handshakes by 2 within a group to avoid double counting and count total number of handshakes between different groups by simple multiplication. Here, it will be   + 10*7 = (90/2) + 70 = 115.

amt88 · Answer

I have a doubt on this question. It says that each business executive shakes the hand of every other B. Executive. From this statement, I understand that each B. Executvies shakes hands with 4 of the other business executives. Can someone explain me why we are considering that all B. Executives shake hands with each other?

Bunuel · Answer

The statement "each business executive shakes the hand of every other business executive" means that each executive shakes hands with the other 9 executives, not just 4. They shake hands with all other executives present, excluding themselves.

DJ1986 · Answer

I understood it the way it was intended but I can see how some people could be confused.

The way I solved was by multiplying execs by chairmen = 70

Then I thought, the first Exec can shake 9 other executives. Then the next one can shake 8, 7,6,5,4,3,2,1 which adds to 45.

70+45 = 115

energetics · Answer

Step 1: Execs shake hands everyone
Ways to choose * Choices = 10 execs * 16 (they don't shake their own hand) = 160

Step 2: chairmen shake hands with execs 
Ways * Choices = 7 * 10 = 70

160+70 = 230, now we divide by 2 to get rid of the extras (there is no difference between person A shaking hands with B or person B shaking hands with A) 230/2 = 115.

soondoobu · Answer

Bunuel  how did you get 10C2 for the number of handshakes between business executives? I am struggling to understand how you got the 2 from 10C2. I appreciate the help.

Bunuel · Answer

2 people engage in a handshake. So, the number of handshakes between 10 people is the number of pairs possible out of 10 people, hence 10C2.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

HenryHai · Answer

Hi,
I have a question that: why do 10 business executive shaking hands with 7 chairmen turning into 10 x 7 = 70, meanwhile 10 business executive we cannot do the same thing?
For example, we have 10 business exe, we divide them into 2 groups: 5 5. Thus, 5 x 5 = 25 can be the number of shaking-hand turns between business executives. Can you please explain this one?
Thank you in advance,
Best regards

Bunuel · Answer

Because dividing the 10 business executives into two groups of 5 omits the handshakes within each group. In each group of 5, the handshakes amount to 5C2 = 10. Adding these to the 25 handshakes between the two groups gives 25 + 10 + 10 = 45. This matches the result of the standard calculation, 10C2 = 45.

vaishnavi2498 · Answer

I have a doubt, why in approach we haven't divided 10* 7 by 2? Are there no repeated handshakes counted?

Bunuel · Answer

Each of the 10 executives shakes hands with each of the 7 chairmen, so it's directly 10 * 7. No need to divide by 2 because there’s no repetition or double counting.

To see this clearly, try picking two small groups, say A, B, C and X, Y, and simply listing all cases. Each of the 3 in the first group shakes hands with each of the 2 in the second group. That's 3 * 2 = 6 handshakes, not 3:

A - X
B - X
C - X
A - Y
B - Y
C - Y

avellanjc · Answer

Bunuel,

I see that your simple example shows that one shouldnt divide by 2 or 2! when counting the handshakes between execs and chairmen. But doesnt the theory of the sloth method for counting says that when one is counting combinations (order doesnt matter.. ab=ba) one should divide the multiplication by n!, where n is the # of sloths (or the # of pickings,decisions being made?

Please give me a clear rationale as to when one should divide.

Thanks in advance,

Juan C. Avellan

Bunuel · Answer

You divide by 2 when the same two people can be chosen in reverse order, like A-B and B-A in one group.

Here, the two groups are different: one side is executives, the other is chairmen. A handshake between an executive and a chairman can happen only once, so there is no duplicate counting. That’s why we do not divide by 2.

M07-27

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