Create your permutation, combination, and probability distribution questions.

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.

Explore a set of permutation, combination, and probability distribution questions designed to enhance your understanding of these mathematical concepts. Includes step-by-step solutions, key formulas, and probability distribution properties. Suitable for students in Grades 10-12.

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