A group consists of a few men and a few women. What is the total numbe

Hi Bunuel
Can you provide the official explanation or point out if there are any errors in my approach because this is a weird question as getting two different answers from each statement which is not possible in a DS question

Statement 1: In a circular arrangement if men & women are sitting in alternate position, then number of men and women have to be equal.
let the number of men = women = \(n\)
\(n\) people can be arranged in a circular table in \((n-1)!\) ways and for the remaining \(n\) people there are \(n\) spaces available now. so the remaining \(n\) people can be arranged in \(n\)! ways.
Thus total number of ways of arranging men and women in a circular table, alternately \(= (n-1)!*n!\)

In a linear arrangement if we start with men and then alternately arrange men and women, then the number of ways will be \(= n!*n!\)
but we can also start with women, hence total number of ways of linearly arranging men & women alternatively \(= 2*n!*n!\)

As per statement \(\frac{(n-1)!*n!}{2*n!*n!}\) = \(\frac{1}{10}\)
Solving the equation we get \(n = 5\). Hence the statement is sufficient

Statement 2: Clearly states that number of handshakes are equal, hence number of men = number of women = \(n\)
Number of handshakes among each men or women themselves \(= n_C_2\)
Total number of handshakes between \(n\) men & \(n\) women \(= n*n = n^2\)
as per the statement \(n_C_2\) = \(\frac{5}{12}*n^2\)
Solving the equation we get \(n = 6\). Hence Sufficient.

Option D

About solution of statement 2:

The total number of handshakes (each man with each woman) = n*n=sqr(n)
When men shake hands with each other, the number of handshakes for first man = n-1
for the second - n-2
...
for the last - 0.
It is arithmetic progression, sum it up.
n-1 + n-2 + ... + 0 = ((n-1 + 0)/2)*n
So ((n-1 + 0)/2)*n = 5/12*n*n --->n = 6.

I would like to know too about different answers.
But I suppose, we put in statement 1 not 1/10, but 1/12, and everything will be ok

Bunuel,
please provide a solution for this :(
thank you very very much

What is statement 2 actually mean? Can anyone translate it for me,

if the men and women shake hands only with each other such that every man shakes hands with every woman, each is equal to 5/12 of the number of handshakes possible , <<what is that mean? What is the 5/12 stand for?

Thank you so much

Hi pclawong

Statement 2 implies that if men shake hands among themselves or if women shake hands among themselves, then that number will \(\frac{5}{12}\) \(of\) the number of handshakes if every men shakes hands with every women.

So for example if the number of handshakes among men themselves is \(x\)
and the number of handshakes among men and women = \(y\)
then \(x= \frac{5}{12}*y\)