Cards and columns?

Question

I have a deck of cards which I arrange in 4 rows (13 columns). If I pick one card from each column, how do I show that it's possible to get one card from each rank? I know that I have to use the Pigeonhole Principle.

Answer 1

If I'm not mistaken, If you just flatten out a shuffled deck in $4$ rows and pick $1$ card from each column, then this is equivalent to picking $13$ random cards from a deck.

I'm not sure why you accepted this answer when Thomas Andrews provided a short explanation involving the pigeon principle you asked about: https://math.stackexchange.com/q/1685770

Answer 2

You probably need pigeonhole along with the Hall's marriage theorem.

Using Hall's theorem, you need to show that any $n$ columns has at least $n$ different ranks. Since $n$ columns contain $4n$ cards, and there are only four cards of each rank, there cannot be fewer than $n$ different ranks in the set. (That is the application of the pigeonhole principle.)