Probability of taking cards in ascending order from a standard deck?

Question

Assume that we have a standard deck of $52$ cards and $J=11, Q=12,$ and $K=13$. We pick $x$ cards from the deck. What is the probability that the cards we picked are in strictly ascending order?

Answer 1

There are $\binom{13}x$ ways to pick $x$ different ranks, $1$ way to order them in strictly ascending order, and $4^x$ ways to choose suits for them, whereas there are $\frac{52!}{(52-x)!}$ ways in total to pick $x$ cards, so the probability is

$$ \frac{\binom{13}x\cdot4^x\cdot(52-x)!}{52!}\;. $$

For small $x$, where the probability of picking the same rank more than once is low, this is roughly $\frac1{x!}$.