In pre-calculus, both permutations and combinations are used to count the number of ways to arrange or select items. The key difference lies in whether the order of selection matters.
A permutation is an arrangement of objects in a specific order. When we calculate permutations, the sequence in which the items are arranged is important. For example, ABC is a different permutation from BAC.
A combination is a selection of objects where the order does not matter. In combinations, we are only concerned with which items are selected, not the order in which they are selected. For example, ABC is the same combination as BAC.
| Feature | Permutations (nPr) | Combinations (nCr) |
|---|---|---|
| Definition | Arrangement of items in a specific order. | Selection of items where order doesn't matter. |
| Order | Order is important. | Order is not important. |
| Formula | $nPr = \frac{n!}{(n-r)!}$ | $nCr = \frac{n!}{r!(n-r)!}$ |
| Notation | Also written as P(n, r) or ${}_nP_r$ | Also written as C(n, r) or ${}_nC_r$ or $\binom{n}{r}$ |
| Number of Outcomes | Generally more outcomes than combinations for the same n and r. | Generally fewer outcomes than permutations for the same n and r. |
| Example | Arranging 3 books on a shelf. | Choosing 3 students from a group of 5 to form a committee. |
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