Handshakes in a party Here's the question which I'm stuck with -

There are $20$ married couples at a party. Every man shakes hands with everyone except himself and his spouse. Half of the women refuse to shake hands with any other women. The other $10$ women all shake hands with each other (but not with themselves). How many handshakes are there at the party?

My Solution:


*

*Handshakes done by men: There are $20$ ways to pick a man and $38$ ways to pick the other person, totaling to $20 \cdot 38 = 760$. But since every handshake is counted twice, the answer is $\frac{760}{2} = 380$

*Handshakes done by the women who refuse to shake hand with any other women: these are already counted in $380$ handshakes (Because the women can only shake hands with men. This was taken care of above).

*Handshakes done by the other $10$ women and men: These are counted in $380$ handshakes. 

*Handshakes done by women and women in the group of $10$: are $\binom{10}{2} = 45$.


Hence the total handshakes are $380 + 45 = 425$.

However, the total handshakes are $615$ according to my textbook. Can anyone please help me to find out the mistake?
 A: There are the following types of handshakes


*

*Woman vs woman: $\dbinom{10}{2}=45$. You were correct on that.

*Man vs man: $\dbinom{20}{2}=190$. 

*Man vs woman: each man performs $19$ handshakes with women. Since there are $20$ men this gives $$19\times20=380$$ such handshakes. 


Summing up $$45+190+380=615$$ Your mistake was in steps $2$ and $3$ because you treated them as $1$ step. While handshakes between man and man are counted indeed twice, handshakes between man and woman are counted only once, so dividing all with $2$ is wrong. You have to distinguish between these two categories.
A: Your mistake can be seen in your first line: you should not divide by $2$ as you did not count the handshakes between men and women twice. 
Instead, the ways to pick a man is $20$. The number of men that shake hands with him is $19$. Since very handshake is counted twice, the men shake hands $190$ times. 
The number of handshakes men had with women is simply $20 \times 19$, thus $380$. 
Combining this with your results gives the answer in the textbook, or $570+45=615$. 
