# Contest Probability - Determining the mass distribution function

Five(5) men and five(5) women patricipate in a contest. They are classified according to their performance and two people cannot take the same place.Obviously,there are 10! combinations of rankings. Let X be the the highest rank taken by a woman (for example, X=1 if a woman took the first position, we have X=6 if the five women ranked in the last positions). Find the mass distribution function of X.

I thought that range is Sx=(1,2,3,4,5) and that Px(x)=P(X=x)= Sx / 10!. Is that correct? If yes can you help me determine Sx? If not can you tell me what methodology should I follow?

Distinguishing only men from women there are $\binom{10}5$ possible arrangements.
Random variable $X$ takes values in $\{1,2,3,4,5,6\}$ and in $\binom{10-k}4$ of these arrangements we have $X=k$.
(If e.g. $X=3$ then the $4$ women with lower rank must be spread over the $10-3=7$ positions $4,5,6,7,8,9,10$)
By equal probability of each possible arrangement: $$P\left(X=k\right)=\binom{10-k}4\binom{10}{5}^{-1}$$