Math Question, total number of handshakes?

Question

If there 17 boy-girl pairs, and in which all the girls shake hands amongst themselves, and every boy shakes hands with only a girl. What is the total number of handshakes?

Answer 1

　

There are 34 people, so the number of handshakes if everyone shook hands with every one else would be: C(34,2)



However, the 17 boys do not shake hands with each other. So we must subtract C(17,2)



C(34,2) − C(17,2) = 561 − 136 = 425





NOTE:

atsuo's method is different than mine but correct

The other 2 answers have a small mistake calculating handshakes among girls.

Answer 2

17 girls

Each girl can shake hands with the other so there are 17x 16 = 272 handshakes. But we'd be counting twice.

Therefore, number of handshakes among the girls is 272/2 = 136.

Each boy shakes hands with each girl. 17x17 = 289 handshakes.



Total = 136+289 = 425

Answer 3

Each of the 17 girls shakes the hand of 16 girls. Multiply those numbers and divide by 2 to avoid counting any handshake twice.



(17)(16)/2 = 136



Each boy shakes hands with exactly one girl. That is 17 handshakes. Each of these handshakes is counted only once, because we are not counting how many boys' hands shaken by the girls.



total number of handshakes = 136 + 17 = 153



My total appears to be an outlier among the responses. The discrepancy may be in the interpretation of this condition: "(E)very boy shakes hands with only a girl." To me, that sentence does not seem ambiguous in the least. Every boy shakes the hand of one girl. That is 17 handshakes, and there is no mention of any handshakes between two boys.



Follow-up:

I am not changing my answer, but now I do have to allow that there is some ambiguity in that one condition. It is given that every boy shakes hands with only a girl. I took that to mean exactly one girl. That is correct literally, but perhaps the intention was only to exclude handshakes between two boys. If that were the case, then we still would not know how many girls' hands were shaken by each boy. I see no reason to assume 17.



This is an ongoing issue with combinatorics problems. It seems that the conditions are never clear enough.

Answer 4

One shakehand happens between two persons .

So the number of shakehands is the number of ways to choose two persons .



But we can not choose two boys .



We can choose one boy out of 17 boys and one girl out of 17 girls .

So 17 * 17 = 289 ways exist .



Or we can choose two girls out of 17 girls .

So 17C2 = 17*16 / 2 = 136 ways exist .



So the total number of handshakes is 289 + 136 = 425 .

Answer 5

Each girl can shake the hands of 16 other girls. So that's 17*16 = 272 hand shakes. And 17 boys shake hands with 17 girls, that 17*17 = 290 hand shakes. So the total is 562 shakes.

Answer 6

a bunch