Factorials, Permutations and Combinations
Counting arrangements and selections without listing them: the multiplication principle, factorials, nPr for ordered arrangements and nCr for unordered selections, plus identical items, circular arrangements and the at-least/at-most split.
What you will be able to do
- Apply the multiplication principle to count outcomes of a multi-stage choice.
- Compute nPr and nCr and say which one a question needs.
- Adjust a count for identical items and for a circular arrangement.
- Split an 'at least' or 'at most' selection into disjoint cases and add them.
What is inside this lesson
- 1Counting without listing
- 2The multiplication principle
- 3Factorial
- 4nPr and nCr
- 5Identical items, circles, and at-least
- 6Worked example
- 7Common mistakes
- 8Quick revision
- 9Practice
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Counting without listing
Some questions ask how many ways something can happen — arrange six people in a row, pick a team of three from ten, form a four-digit code. Listing every case is hopeless under time pressure, so we count with rules instead.
Two questions decide which rule to use:
- Does order matter? "First, second, third" and "a code" care about order — that is a permutation. "A committee", "a handshake", "a selection" do not — that is a combination.
- Are the items all different? Repeated letters or identical balls change the count.
Get those two right and the formula follows.
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