# How many shake hands and hugs were exchanged at that party? [closed]

There are 20 people at this birthday party, where 10 are men, 5 kids and rest women. So, the etiquette here is like follows:

a. when men meet men for the first time, they greet by shaking hands

b. when men meet women for the first time, they greet with a hug.

c. when women meet women for the first time, they too shake hands with each other.

d. Every Adult greet the birthday boy (man) with both shake hands and hugs.

d. kids just say "hi". No shake hands, no hugs.

So, My question is, How many shake hands and hugs were exchanged at that party?

• Is mention of ...first time a must ? – Mea Culpa Nay Oct 10 '17 at 13:20
• I think that specifies a "once per pair" relationship, as there's no mention of subsequent meetings. – Chris Cudmore Oct 10 '17 at 13:21
• some men could be kids, that means there are 10 men, 10 women and 5 kids? :) – Oray Oct 10 '17 at 14:06

$60$ handshakes and $59$ hugs are performed, assuming everyone meets each other for the first time.

The birthday boy shakes and hugs with each other man and woman:

$14$ shakes, $14$ hugs

The remaining $9$ men shake hands with each of the other (non-birthday) men:

$\left(\array{9 \\ 2}\right) = \frac{9*8}{2} = 36$ shakes

The $5$ women shake hands with each of the other women:

$\left(\array{5 \\ 2}\right) = \frac{5*4}{2} = 10$ shakes

The $9$ men each hug the $5$ women:

$9*5=45$ hugs

The kids neither handshake nor hug anyone, so they can be ignored.

If we follow the assumption that 'men' and 'women' are by definition adults, and the remaining are adults (so 10 adult men M, 5 adult women W and 5 kids K, no tricks involved), we have 6 combinations:

MM / MW / MK / WW / WK / KK.

For MM: 10 Men have 45 encounters amongst eachother. (n x (n-1))/2 This leads to 45 handshakes

For MW: 10 men and 5 women imply 50 encouters. This leads to 50 hugs

For MK: 10 men shake hands and hug the birthday boy (so only one kid interacts, the others just say hi). This leads to 10 handshakes and 10 hugs.

For WW: 5 women shaking hands with eachother leads to 10 handshakes (5 x 4)/2, similar as with the men.

For WK: 5 women shake hands and hug the birthday boy (so only one kid interacts, the others just say hi). This leads to 5 handshakes and 5 hugs.

For KK: No handshakes or hugs.

If we make the sum, we get 70 handshakes and 65 hugs.

• The birthday boy is stated to be a man, not a kid. – T. Linnell Oct 10 '17 at 14:24
• You're right. It's just poor wording though, and basically this seems like a math homework (and voted as such). The logic is the similar. Given the VTC passed, I'll not update the answer. (and well, basically you typed it up like that anyway) – Tim Couwelier Oct 11 '17 at 7:15