Probability of drawing particular card first

Question

Probability of drawing a card from a deck of $52$ cards is $1/52$. But if I want to calculate a probability of drawing two cards where first one is a specific one (e.g. Ace of Clubs).

So there are $52!/(2!(52-2)!) = 1326$ ways to draw two cards. There are $\textbf{51}$ way to draw two cards where first one - is a Ace of Clubs.

So why $51/1326$ doesn't equal to $1/52$? Where am I wrong?

Answer 1

When you talk of ace of clubs being first, you are considering the order in which cards are being drawn.

In general, there will thus be $52\times51$ ways in which two cards can be drawn.

Against this, drawing ace of clubs followed by some other has $1\times 51$ ways,
which resolves your anomaly.

Answer 2

For the sample space $52\cdot51$, we ask the question

How many outcomes start with the Ace of Clubs, answer $51$.

For the sample space $\binom{52}{2}$, we ask the question

How many outcomes contain the Ace of Clubs, answer $51$.

They are the same.