Counting techniques

To improve the security of a mobile app, a developer decides to create access codes consisting of an uppercase letter (from the standard 26-letter alphabet) followed by three digits (from 0 to 9).

• How many different codes can be created if digits can be repeated?

• How many different codes would there be if the three digits must be distinct from one another?

• If a code with repeatable digits is generated at random, what is the probability that the letter is a 'Z'?

To improve the security of an esports platform, users are required to create a 6-character password. The first two characters must be uppercase letters of the Spanish alphabet (27 letters) and the remaining four must be digits from $0$ to $9$.

Calculate how many different passwords can be generated if:

1. a) Repetitions of letters and numbers are allowed.
2. b) Repeating letters is not allowed, but repeating numbers is allowed.
3. c) The letters must be distinct and the numbers must form a strictly increasing sequence (for example: $1258$, but neither $1225$ nor $4321$).

A cybersecurity company generates access keys by permuting the characters of the word BLOCKCHAIN. One of these keys is chosen at random.

1. a) Calculate the probability that the chosen key starts and ends with a vowel.
2. b) Determine the probability that all the vowels appear together in the key.
3. c) Given that the key starts with B, what is the probability that the second letter is an L?

An engineering class has to choose a committee of 3 representatives. If there are 20 students in the class, calculate:

• In how many ways can the committee be chosen if the three roles (class representative, deputy representative, and secretary) are distinct?

• In how many ways can it be chosen if the three representatives have equal rank and order does not matter?

• If there are 12 men and 8 women, what is the probability that the committee (without distinct roles) consists entirely of women? (Express the fraction using combinations).

An urn contains 12 glass balls for an optics experiment: 5 are blue, 4 are red, and 3 are yellow. If we draw a ball at random, calculate the following probabilities using Laplace's rule:

• That the ball is red or yellow.

• That the ball is not blue.

• That the ball is white.

In a robotics laboratory, 8 proximity sensors in perfect working order and 2 defective sensors have been mixed together in a box. A student draws two sensors consecutively and without replacement (without returning the first one to the box).

Calculate the probability that:

• Both sensors work correctly.

• The first is defective and the second is in good working order.