The gap between them is exactly 3!, and once you can see why, the two formulas stop being a coin flip.
Choosing 3 letters from A B C D E. The left column is the distinct choices. Fan them out into every arrangement.
Ten choices. Now ask whether the order matters.
Does swapping two of them give a different outcome? If yes, order matters and it is a permutation. If no, it is a combination. A committee of three is a combination; a first, second and third place is a permutation of the same three people.
The product rule underneath it all: if one thing can happen in a ways and another in b ways, together they happen in a × b ways. 3 starters, 4 mains and 2 puddings give 24 meals, and every counting formula is this rule applied carefully.
| Order matters | Order does not | |
|---|---|---|
| repeats allowed | nr: 10⁴ = 10000 PINs | beyond this course |
| no repeats | nPr: 5040 | nCr: 210 |
At Higher Level the theorem is extended to fractional and negative indices, where the expansion never terminates.
It only works for |x| < 1. With a positive whole-number index the expansion stops and is exact everywhere. With a fractional or negative one it runs for ever and converges only inside that range, so stating the condition is part of the answer.
1. Find ⁵P₃.
2. A committee of 3 is chosen from 10 people. How many committees?
3. A 4-digit PIN, digits 0 to 9, repeats allowed. How many?
Saying which it is and why, in one line, before any numbers. "Order matters because the positions are different" is the mark.
Multiplying independent stages rather than adding them.
On the extended binomial, stating the range of x for which the expansion is valid.
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