# Probability of two teams from same league level avoiding each other in cup draw

There was a football cup draw this week in my home country. There were 32 teams in the draw, where 16 are from the top league level, and 16 from the second level.

All matches in the draw happened to include one team from the top level, and one from the second level, ie no team from the same level meet each other. It seems a very unlikely event, and I have been trying to calculate the probability of it happening, but I am not able to do so.

Could anyone help me?

In the first match, the probability that the away team is in a different league from the home team is $\frac{16}{31}$ (because after picking the home team, there are $31$ teams left, of which $16$ are from the other league).
In the second match, the probability that the away team is in a different league from the home team is $\frac{15}{29}$ (because after picking the home team, there are $29$ teams left, of which $15$ are from the other league).
In the last match, the probability that the away team is in a different league from the home team is $\frac{1}{1}$ (because after picking the home team, there is $1$ team left, which is necessarily from the other league.
Now just multiply these $16$ probabilities together. What do you get? (It is indeed an unlikely event $-$ about $1$ in $10000$.)