What is wrong with the following solution to a basic combinatorial type probability question?

I am reviewing my probability lecture notes and I decided to try and do the exercises that were solved in the lecture.

I tried to solve the following problem, but came up with a different answer than the one given in the lecture, so I suspect I got it wrong.

We draw cards from a deck of cards (with $52$ cards), what is the probability that the first king was drawn at the $n-th$ draw ?

My attempt:

The total number of sequences of $n$ cards is $\binom{52}{n}\cdot n!$ .

There are $\binom{4}{1}$ ways to get the king in the $n-th$ draw.

There are $\binom{52-4}{n-1}\cdot(n-1)!$ sequences of $n-1$ cards with no king in them.

Hence the answer is $$ \frac{\binom{4}{1}\cdot\binom{48}{n-1}\cdot(n-1)!}{\binom{52}{n}\cdot n!} $$

Can someone please help me understand my mistake ?

via Recent Questions - Mathematics - Stack Exchange http://math.stackexchange.com/questions/294428/what-is-wrong-with-the-following-solution-to-a-basic-combinatorial-type-probabil