Textbook Problem

Find the error in the following reasoning: “Consider forming a poker hand with two pairs as a five-step process.”
Step 1: Choose the denomination of one of the pairs.
Step 2: Choose the two cards of that denomination.
Step 3: Choose the denomination of the other of the pairs.
Step 4: Choose the two cards of that second denomination.
Step 5: Choose the fifth card from the remaining denominations.
There are $\left(\begin{array}{c}13\\ 1\end{array}\right)$ ways to perform step 1, $\left(\begin{array}{c}4\\ 2\end{array}\right)$ ways to perform step 2, $\left(\begin{array}{c}12\\ 1\end{array}\right)$ ways to perform step 3, $\left(\begin{array}{c}4\\ 2\end{array}\right)$ ways to perform step 4, and $\left(\begin{array}{c}44\\ 1\end{array}\right)$ ways to perform step 5. Therefore, the total number of five-card poker hands with two pairs is $13\cdot 6\cdot 12\cdot 6\cdot 44=\mathrm{247.104.}"$