Question

In how many ways can a 5-card poker hand be void (i.e. have no cards in) at least 1 suit?

Answer 1

`575,757`

Explanation:

Out of a 52 card deck, we're only going to allow ourselves to have hands involving 3 (or less suits) - which means we'll only allow hands using `52-13=39` cards - or the cards of 3 suits.

This gives combinations from 39 cards and we pick 5:

`C_(n,k)=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

`C_(39,5)=(39!)/((5!)(39-5!)`

`(39!)/(5!34!)=(39xx38xx37xx36xx35xx34!)/(34!xx120)=575,757`