Here Is an Incorrect Solution to a Problem $\left. \begin{array} { l } 4 \\ 2 \end{array} \right)$

question 152

Essay

Here is an incorrect solution to a problem. Find the error, explain why it is not correct, and give the correct answer. "Problem: Find the number of ways to get two pairs of two different ranks (such as 2 jacks and 2 fives) in a 4-card hand from an ordinary deck of 52 cards."
"Solution: There are 13 ways to get a rank (such as "kings") for the first pair and ( $\left. \begin{array} { l } 4 \\ 2 \end{array} \right)$ ways to get a pair of that rank. Similarly, there are 12 ways to get a rank (such as "sevens") for the second pair and $\left( \begin{array} { l } 4 \\ 2 \end{array} \right)$ ways to get a pair of that rank. Therefore there are $13 \cdot \left( \begin{array} { l } 4 \\ 2 \end{array} \right) \cdot 12 \cdot \left( \begin{array} { l } 4 \\ 2 \end{array} \right)$ ways to get 2 pairs."

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