Here are some original questions covering permutations, combinations, and probability distributions:

Permutation Questions:

1. How many ways can 5 people be arranged in a line if two of them must always be next to each other?
2. In how many different ways can the letters of the word "MATRIX" be arranged?
3. How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 without repetition?
4. A class has 10 students. How many ways can the teacher assign 1st, 2nd, and 3rd prizes to them?
5. You have 6 books, but you can only fit 3 on your bookshelf at a time. How many different ways can you arrange 3 books out of 6?

Combination Questions:

1. In how many ways can a committee of 4 people be chosen from a group of 10?
2. A box contains 12 red balls, 8 blue balls, and 5 green balls. How many ways can 5 balls be selected if at least one must be green?
3. In how many ways can you select 3 fruits from a basket of 6 apples, 4 oranges, and 3 bananas?
4. From a deck of 52 cards, how many ways can 5 cards be drawn such that at least one card is a heart?
5. How many ways can a basketball team of 5 players be selected from 9 players if two specific players must be on the team?

Probability Distribution Questions:

1. A die is rolled 3 times. What is the probability distribution of the sum of the numbers rolled?
2. The probability that a bulb lasts more than 1000 hours follows an exponential distribution with a mean of 1200 hours. What is the probability that a bulb lasts more than 1500 hours?
3. A company makes lightbulbs where 5% of them are defective. If a sample of 10 bulbs is taken, what is the probability distribution for the number of defective bulbs?
4. In a game, a spinner is divided into 5 equal sections numbered 1 to 5. If the spinner is spun 3 times, find the probability distribution for the sum of the numbers that appear.
5. A discrete random variable $X$ takes values 0, 1, 2 with probabilities 0.2, 0.5, and 0.3, respectively. Find the probability distribution of $2X + 1$.

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Would you like details or clarifications on any of these? Here are 5 related questions to expand your understanding:

1. What is the difference between a permutation and a combination?
2. How does the number of outcomes change if repetition is allowed in permutations?
3. Can a probability distribution function have values greater than 1?
4. How do you calculate the expected value of a probability distribution?
5. What conditions must hold for a binomial distribution to apply?

Tip: In permutation problems, if objects must remain together, treat them as a single object first, then calculate permutations.