# Permutations vs Combinations

### Mixed List of Word Problems on Permutations and Combinations

#### Problem 1: Choosing a President and Vice President
Your club is electing a president and a vice president from among 5 members. How many different ways can these positions be filled?

- **Type**: Permutation (Order matters: choosing a president first and then a vice president results in a different outcome than vice versa.)

#### Problem 2: Creating Passwords
How many unique 4-digit passwords can be created using the numbers 0 through 9 if each number can only be used once?

- **Type**: Permutation (Order matters: 1234 is a different password than 4321.)

#### Problem 3: Bookshelf Arrangement
You have 4 different books and you want to arrange 2 of them on a shelf. How many different arrangements are possible?

- **Type**: Permutation (The order in which the books are arranged matters.)

#### Problem 4: Selecting a Committee
From a group of 8 people, you need to select a committee of 3, regardless of the position. How many ways can you select this committee?

- **Type**: Combination (Order does not matter: selecting person A, B, and C is the same as selecting C, A, and B.)

#### Problem 5: Tournament Rankings
In a chess tournament, how many ways can the 1st, 2nd, and 3rd places be awarded among 8 competitors?

- **Type**: Permutation (Order matters: who comes in 1st, 2nd, or 3rd changes the outcome.)

#### Problem 6: Designing a Garden
You are designing a garden and have space for 4 different types of flowers out of 10 varieties you like. How many different selections of flowers can you make?

- **Type**: Combination (Order does not matter: selecting roses, tulips, daisies, and sunflowers is the same as selecting daisies, tulips, roses, and sunflowers.)

#### Problem 7: Seating Arrangements
How many ways can 5 people be seated around a circular table?

- **Type**: Permutation (Order matters, but this is a circular permutation: one needs to account for rotations being considered the same arrangement.)

#### Problem 8: Science Fair Projects
A science fair committee needs to choose 3 projects to represent the school at a national competition from 10 projects that won at the school level. How many ways can the selection be made?

- **Type**: Combination (Order does not matter: selecting projects A, B, and C is the same as selecting C, A, and B.)

#### Problem 9: Creating a Security Code
A security system requires a 6-symbol code, where each symbol can be a letter (26 options) or a digit (10 options). If symbols can be repeated, how many different codes can be created?

- **Type**: Permutation (Order matters, and repetition is allowed, increasing the complexity.)

#### Problem 10: Jury Selection
From a pool of 30 potential jurors, a 12-person jury needs to be selected. How many different juries can be formed?

- **Type**: Combination (Order does not matter: the focus is on who is selected, not the order of selection.)

#### Problem 11: Music Playlist
You are creating a playlist with 7 different songs out of 15 that you have selected. If the order in which the songs are played matters, how many unique playlists can you create?

- **Type**: Permutation (The sequence in which the songs are played changes the playlist.)

#### Problem 12: Math Competition Team
A school can send a team of 4 students to a math competition. If there are 12 students interested and the team needs a leader, a deputy leader, and two members, how many different teams can be formed?

- **Type**: This problem involves both permutation (selecting a leader and deputy leader) and combination (selecting team members). It requires a combination to select 4 students from 12, then a permutation to assign the leader and deputy leader roles among those 4.

These problems are designed to progressively challenge students to understand and apply the concepts of permutations and combinations, enhancing their problem-solving skills and mathematical reasoning.