Probability of drawing hearts 2 after hearts 3

Question

What is the probability of drawing Hearts 2 after you have picked Hearts 3?

My teacher gave this solution: 51*50!/52!

But I don't get how he came to this and which formula he used.

Answer 1

There are $52!$ different sequences of cards. Some of them show the pattern you're looking for, others don't.

The 3H can be drawn as the 1st card, up to the 51st card. That makes 51 possibilities. The 2H is drawn immediately afterwards and doesn't add to the number of possibilities.

The remaining 50 cards can be distributed over the remaining 50 places, giving $50!$ sequences.

Putting this together, you have $51 * 50!$ different matching patterns out of $52!$ possible patterns. Hence the probability of drawing a 2H immediately after a 3H is $$ \frac{51 * 50!}{52!} $$

Answer 2

There are $52!$ ways to order a deck with 52 cards.

How many of them include the segment $3$ hearts --$2$ hearts?

By considering these two cards as one, there are $51!$ orderings that include such sequence.

Thus the probability of having it is $51!/52!=1/52$.

Look, teacher was right!