I am trying to solve the following problem in basic probablity theory :
We are randomly picking $ 6 $ balls numbered $1 - 50$ (out of $50$, without replacment). If $X=\textit{minimum number}$ and $Y=\textit{maximum number}$ find the function $f_{X,Y}(k,l)=P(X=k,Y=l)$.
My approach is the following:
First, suppose that $0<k<l<=50$ and $k<=l+5$. Then: $P(X=k,Y=l)=\frac{\textit{favourable}}{\textit{all}}=\frac{{l-k-1}\choose{4}}{50\choose 6}$.
Now, in my notes instead of the above expression I have :$\frac{6\cdot5 {{l-k-1}\choose{4}}}{50\choose 6} $ but this seems incorrect , mainly because if instead of $50$ balls we had,say, $7$ the latter formula is wrong.
I would appriciated telling me if I am correct or pointing out my mistake(s).
As has already been stated in the comments, you're right and the notes are wrong. Your numerator and denominator both count the number of selections without order, whereas the numerator in the notes counts the number of ways of drawing the six numbers e.g. with $k$ being drawn first and $l$ being drawn last.