Using the digits from 1 to 5 and a multiplication sign $\times$ for $\times 12$, Clara forms the product of two numbers, with the sign $\times$ between them. How should Clara place the cards to obtain the largest possible product?

$\begin{array}{lll}3 & 4 & 5\end{array}$
Observe that we obtain the largest possible result if one of the numbers starts with the digit 5 and the other with 4. Additionally, since we only have five digits, one of the two numbers must have only one or two digits. Let's look at the possibilities that give the largest product.

- one of the factors has one digit:

$5321 \times 4=21284 ; 4321 \times 5=21605$.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-090.jpg?height=162&width=434&top_left_y=690&top_left_x=1439)

- one of the factors has two digits:

$$
\begin{aligned}
& 532 \times 41=21812 ; 531 \times 42=22302 ; 521 \times 43=22403 \\
& 432 \times 51=22032 ; 431 \times 52=22412 ; 421 \times 53=22313
\end{aligned}
$$

Therefore, the best result is $431 \times 52=22412$.