Here's the full text of the problem: "Draw from a deck without replacement. Find the probability that the $10^{th}$ card is a king and the $11^{th}$ is a non-king."
The text gives the answer as "$P$(10th is king, 11th is non-king) = $P$(1st is king, 2nd is non-king) $=\frac{4}{52}\cdot\frac{48}{51}$". But doesn't this assume that no kings had been drawn by the 10th draw? I don't see any justification for that in the problem statement.
My attempt at a solution approached this as two-stage problem. The first stage has four possible starting states: 1) 10 card hand with 1 king; 2) 10 card hand with 2 kings; 3) 10 card hand with 3 kings; 4) 10 card hand with 4 kings. The second stage, or 11th card draw, has two possible outcomes: king or non-king. So I applied the total theorem of probability to get:
\begin{align} P(\text{non-king}) &= P(\text{10 cards drawn with 1 king})P(\text{11th draw is non-king|10 cards drawn with 1 king}) + P(\text{10 cards drawn with 2 kings})P(\text{11th draw is non-king|10 cards drawn with 2 kings}) + P(\text{10 card hand with 3 kings})P(\text{11th draw is non-king|10 cards drawn with 3 kings}) + P(\text{10 card hand with 4 kings})P(\text{11th draw is non-king|10 cards drawn with 4 kings}) \end{align} Am I on the right track, or have I made this problem into something more complex than it is? Any help resolving this would be greatly appreciated!