Why does reversing a pair create a different permutation but not a different combination?
Conditions
- The elements being arranged are distinct.
- We are comparing ordered arrangements (permutations) with unordered selections (combinations).
Reasoning, step by step
- Identify that a permutation records the exact order of the selected elements.
- Recognize that swapping the positions of two elements changes this recorded order.
- Identify that a combination records only which elements are selected, ignoring their order.
- Conclude that swapping elements leaves the selected set unchanged, so the combination remains the same.
Example
Choosing two distinct letters from four gives 12 ordered pairs but only six unordered pairs. Reversing a pair creates a different permutation, not a different combination.
Common misconceptions
- Believing that combinations and permutations are identical if the elements are the same.
- Thinking that the order of selection matters in a combination problem.
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Related questions
For selection without repetition, the number of combinations is calculated using the binomial coefficient formula . This formula counts the number of ways to choose distinct elements from a set of elements where order does not matter.
Conditions: The selection is made without repetition.; The elements are distinct.; Order does not matter in the selection.
To calculate the number of combinations with repetition, where order is ignored and repetition is allowed, you use the formula . This counts the number of ways to select objects from distinct types.
Conditions: Repetition is allowed.; Order does not matter.; There are distinct types of objects.; You are selecting exactly objects.
We divide by because the permutation count includes all possible internal orderings of each selected group. Since combinations are unordered selections, these internal orderings are redundant.
Conditions: The selection is made without repetition.; Order does not matter in the final count.
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