Question

How can you tell whether a counting problem is a permutation or a combination problem? Give examples to support your answer...

Answer 1

[AI-generated answer — Bot GPT 3.5]

Permutation and combination problems involve counting the number of ways in which objects can be arranged or selected. The key difference between the two is whether the order of the objects matters or not.

Permutation problems involve arranging objects in a specific order, while combination problems involve selecting objects without considering the order.

For example, consider the following scenarios:

1. Permutation problem: In how many ways can 3 different books be arranged on a bookshelf?
In this case, the order of the books on the shelf matters, so this is a permutation problem. The number of ways to arrange the books can be calculated using the permutation formula: P(n, r) = n! / (n-r)!. In this case, there are 3 books to arrange, so the number of ways is 3! = 6.

2. Combination problem: In how many ways can a committee of 2 people be chosen from a group of 4 people?
In this case, the order in which the committee members are chosen does not matter, so this is a combination problem. The number of ways to choose a committee can be calculated using the combination formula: C(n, r) = n! / [r!(n-r)!]. In this case, there are 4 people to choose from and the committee size is 2, so the number of ways is 4! / [2!(4-2)!] = 6.

In summary, if the order of objects or items matters, it is a permutation problem. If the order does not matter, it is a combination problem.

Answer 2

use simpler terms

Answer 3

[AI-generated answer — Bot GPT 3.5]

Permutation problems are about arranging items in a specific order, where the order matters. Combination problems are about selecting items without considering the order.

For example:
- Permutation problem: How many different ways can you arrange 3 colored balls in a row?
- Combination problem: How many different ways can you choose 2 ice cream flavors from 4 available flavors?

If you need to arrange items in a specific order, it's a permutation problem. If you just need to select items without worrying about the order, it's a combination problem.