Question

EXPLAIN the difference between a combination and a permutation.

Answer 1

The primary difference between a combination and a permutation lies in the importance of the order of elements.

### Permutation:
- **Definition**: A permutation is an arrangement of items in a specific order. The order matters.
- **Example**: For the items \(A\), \(B\), and \(C\), the permutations would include \(ABC\), \(ACB\), \(BAC\), \(BCA\), \(CAB\), and \(CBA\). Each different arrangement counts separately because the order is different.
- **Formula**: The number of permutations of \(n\) items taken \(r\) at a time is given by the formula:
\[
P(n, r) = \frac{n!}{(n-r)!}
\]
where \(n!\) (n factorial) is the product of all positive integers up to \(n\).

### Combination:
- **Definition**: A combination is a selection of items without regard to the order in which they are arranged. The order does not matter.
- **Example**: For the items \(A\), \(B\), and \(C\), the combinations of selecting 2 items would include \(AB\), \(AC\), and \(BC\). The combination \(AB\) is considered the same as \(BA\).
- **Formula**: The number of combinations of \(n\) items taken \(r\) at a time is given by the formula:
\[
C(n, r) = \frac{n!}{r!(n-r)!}
\]

### Summary:
- **Order Matters**: Permutations > Order is important.
- **Order Doesn’t Matter**: Combinations > Order is irrelevant.

This distinction is important in fields such as statistics, probability, and combinatorial mathematics.