Students Ana Beatriz and Paula decided to register on a website with exercises to study for upcoming simulations, but to register on this website, they need to choose a password consisting of five characters, three numbers and two letters (capital letters). or lowercase). Letters and numbers can be in any position. They know that the alphabet is made up of twenty-six letters and that an uppercase letter differs from a lowercase letter in a password. What is the total number of possible passwords for registering on this site?

To find the total number of possible passwords, we need to calculate the number of ways we can arrange the characters.

First, let's calculate the number of ways to choose the position for the numbers. Since we need to choose three numbers out of the five positions, this can be done in $${5 \choose 3}$$ ways.

Next, let's calculate the number of ways to choose the position for the letters. Since we need to choose two letters out of the remaining two positions, this can be done in $${2 \choose 2}$$ ways.

Finally, for each position that has a number, we have 10 choices (0-9), and for each position that has a letter, we have 52 choices (26 lowercase letters + 26 uppercase letters).

Therefore, the total number of possible passwords is:

$$ {5 \choose 3} \times {2 \choose 2} \times 10^3 \times 52^2$$

Simplifying this expression, we get:

$$\frac{5!}{3!\cdot(5-3)!} \times \frac{2!}{2!\cdot(2-2)!} \times 10^3 \times 52^2$$
$$\frac{5!}{3!\cdot2!} \times 1 \times 10^3 \times 52^2$$
$$\frac{5 \times 4}{2} \times 1 \times 10^3 \times 52^2$$
$$10 \times 1 \times 10^3 \times 52^2$$
$$10 \times 10^3 \times 52^2$$
$$10^4 \times 52^2$$

Calculating this expression, we find:

$$10^4 \times 52^2 = 27040000$$

Answer: The total number of possible passwords for registering on this site is 27,040,000.