How many ways can 4 out of 7 candidates be ranked for 4 different awards?

A robotics club has five autonomous bots, each programmed to monitor one of five different zones in a testing facility. If each bot is randomly assigned to a different zone (with no overlaps), how many distinct monitoring assignments are possible?

A certain state’s license plates consist of a sequence of $2$ letters followed by $3$ digits. Each letter can be any of the $26$ English letters, and each digit can be any of the $10$ digits from $0$ to $9$. Letters and digits may be repeated.

For example, valid license plate codes include $AB123$, $ZZ999$, and $QW007$. How many different license plate codes are possible?

A lab technician must randomly select $2$ different chemicals from a shelf containing $13$ labeled samples for a compatibility test. In how many different ways can the technician choose $2$ chemicals?

(Note: The order in which the chemicals are selected does not matter; only the pair of selected chemicals is important.)

A batch of $200$ lightbulbs includes exactly $3$ that are defective. If a quality inspector randomly selects $3$ lightbulbs without replacement, what is the probability that at least one defective bulb is found in the sample?

A hand of $5$ cards is dealt from a standard $52$-card deck. What is the probability that your hand contains exactly $1$ pair, with the other $3$ cards all of different ranks and different from the pair?

A school has $180$ students in grade $12$, consisting of $120$ day scholars and $60$ boarders. The school wants to select $4$ students to form the student council planning committee.

Use technology or a calculator to find the number of ways to choose $4$ students from the $180$.

If you toss a fair coin $6$ times, what is the probability that tails appear exactly twice?

A professor is selecting a panel of $6$ students from a class of $15$ to participate in a debate. How many different panels can be formed?

A password consists of 4 letters (A-Z) where letters can repeat. How many different passwords are possible?