Ref: Disney Japan
Cinderella is at a ball with 49 other people. While dancing, they form 25 pairs (with 2 people in each pair). However, Cinderella really wants to avoid her evil stepmother and 2 ugly stepsisters (total of 3 people), and does not want to dance with any of them.
If there are \(N\) ways to form 25 pairs for dancing in which Cinderella doesn't have to dance with any of her stepfamily, what are the last 3 digits of \(N\)?
Details and assumptions
There are no gender restrictions for the dancing pairs.
The order of the pairs does not matter.
Calculators are allowed for this question. I'm not that mean ^_^.