Work analysis: A teacher picks pairs of students randomly; there are 20 students. What is the probability that Alex and Ali are paired?

Question

anonymous · Answer

$$\frac{\left(\begin{matrix}20 \ 2\end{matrix}\right)-\left(\begin{matrix}18 \ 2\end{matrix}\right)}{\left(\begin{matrix}20 \ 2\end{matrix}\right)}$$

anonymous · Answer

It's not right; the answer is 1/19. Where did I go wrong?

anonymous · Answer

1/10   
because there are totally 20 /2 =10 paris and Alex and Ali are paired can be paired in 1 way

anonymous · Answer

There are 19 other students; Ali could be with any of them. Alex is just one of them, so 1/19. I am wondering why my first method didn't work.

anonymous · Answer

lol...i cant find a way to explain it but maybe this is a clue for u and others...

$$\frac{\left(\begin{matrix}10 \ 1\end{matrix}\right)}{\left(\begin{matrix}20 \ 2\end{matrix}\right)}$$