# What is the probability that each card will have a different number when 5 cards are dealt?

If we choose 5 cards from deck of 52, and card numbers go from 1 to 13... What is the probability that each card will have different number?

Well, you know the sample space is 52 choose 5. Count the number of $5$ card hands that all have different numbers. You have a set of $13$ numbers, and you must choose $5$. Then, each card may take on one of four suits, so multiply $13$ choose $5$ by $4^5$, to obtain the numerator.
$$\frac{52*48*44*40*36}{52*51*50*49*48}$$ The first draw is valid with probability $\frac{52}{52}$, afterwards there are $52-4$ valid cards of $51$ total remaining etc.