Binary Magic Card

I think I first saw this in Ravi Vakil's book. Each card denotes a particular bit. So if there are 3 cards then we can have 8 different numbers. If there are 10 cards we can have all 1024 numbers. In this case we have chosen 5 cards thus all the days of the month are included. The trick works because binary representation of each number is unique. Look at example for number 13.