I'm trying to figure out the probability of drawing a poker hand of no value from a deck of 53 cards (1 additional joker) the context that it cant contain a joker, and none of the cards can be the same number and or face, so this excludes straights flushes and suit patterns.
I think it is simply
$$\frac{^{13}C_5 \times \left(^4C_1\right)^5}{^{53}C_5}$$
Although I am most likely mistaken
$$\frac{\left(\binom{13}{5}-10\right)\times(4^5-4)}{\binom{53}{5}}$$ The $-10$ excludes straights, the $-4$ excludes flushes.