The textbook also states the following:
There are $13$ different kinds of cards, with four cards of each kind.
(Among the terms commonly used instead of “kind” are “rank,” “face
value,” “denomination,” and “value.”) These kinds are twos, threes,
fours, fives, sixes, sevens, eights, nines, tens, jacks, queens,
kings, and aces. There are also four suits: spades, clubs, hearts, and
diamonds, each containing 13 cards, with one card of each kind in a
suit.
which describes it what it means by a 'kind'.
So far I did the following:
$$\Large\frac{\binom{13}{5}}{\binom{52}{5}}$$
I think I am missing something in the numerator however. Maybe $\binom{47}{5}$? But I am not sure how to justify it.
Edit: Had to edit the question, so if somebody already started answering please check it.
Best Answer
We must select five of the $13$ kinds, which can be done in $\binom{13}{5}$ ways. For each of the five kinds we can select, there are four suits from which we can select a card of that kind. Hence, the number of ways we can select a hand in which each card is of a different kind is $$\binom{13}{5} \cdot 4^5$$ Hence, the probability that a five card poker hand contains cards of five different kinds is $$\frac{\binom{13}{5} \cdot 4^5}{\binom{52}{5}}$$