What is the formula for the number of ordered r-item selections from n distinct items without repetition?
Conditions
- The items are distinct.
- Selection is without repetition.
- The order of selection matters.
- is a non-negative integer less than or equal to .
Reasoning, step by step
- Identify the total number of distinct items, .
- Identify the number of items to select, .
- Recognize that the first choice has options, the second has , and so on.
- Multiply these decreasing counts: .
- Express this product compactly using factorials as .
Example
A four-letter password without repeated letters gives possibilities. In general, ordered r-item selections from n distinct items number n!/()!.
Common misconceptions
- Using which allows repetition.
- Using which assumes selecting all n items.
- Confusing this with combinations where order does not matter.
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