I draw a hand of 13 from a deck of 52 standard playing cards. What is the probability that I do not have a card from every suit?
I count the number of ways I can draw 13 from 3 suits
but I mind the intersection. Each possible pair of suits that I may have drawn from only is counted twice. And in the possibility that I pick from only one suit: each possibility is counted three times.
This removes the overcounted iterations from the pair of suits I could have drawn from, but now I'm not considering the possibility that I drew from only one suit, so:
which is the probability I'm looking for.
Is my reasoning sound? Have I made any mistakes? Is there a better solution?