Permutations and Combinations
Count arrangements where order matters and selections where it does not, and see the second as the first divided by the arrangements you refuse to distinguish.
What a learner can do afterwards
- Choose between a permutation and a combination count from the wording of a problem
- Derive the combination formula from the permutation count by dividing out repeats
- Count arrangements of a multiset where some objects are identical
1 · Read
A permutation is a list where order matters: medals, passwords, and officer posts all change when you reorder them. A combination is a group where order does not matter: committees, card hands, and toppings stay the same however you list them. Watch the wording: ranked, arranged, or seated in a row shouts permutation, while chosen or selected whispers combination.
Ten students run for president, vice president, and secretary, three different posts. Anyone of the 10 can take the first post, then 9 remain for the second, then 8 for the third. The count is 10 times 9 times 8, or 720 ordered selections.
A combination starts from that permutation count and divides out the repeats. Each chosen group was counted once per rearrangement, and 3 people rearrange in 3 times 2 times 1, or 6 ways. So a 3-person committee from 10 students is 720 divided by 6, or 120. Symmetry helps: choosing 5 books to shelve from 7 equals choosing the 2 to leave out, so C(7, 5) equals C(7, 2), which is 21.
Identical objects need the same treatment: divide by each repeat factorial. The word LEVEL has 5 letters with E twice and L twice, so its arrangements are 5 factorial divided by 2 factorial times 2 factorial, or 120 divided by 4, which is 30.
Permutations count orders, combinations divide the order away, and repeats divide again.
2 · Watch
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