The factorial between nCr and nPr, made visible.
Ask how many ways there are to choose 3 from 5 before fanning anything out.
Answers split between 10 and 60, which is the lesson. Both are right, to different questions. Fanning each choice into its six orderings shows the 3! that separates them.
The question to ask is whether swapping two of them changes the outcome. A committee says no, a podium says yes, and that one sentence replaces guessing between two formulas.
Three starters, four mains, two puddings give 24 meals. nPr is that rule applied to shrinking choices, and nCr is nPr with the orderings divided out. Neither is a new idea.
On the extended binomial, state |x| < 1. With a positive integer index the expansion terminates and is exact. With a fractional or negative one it runs for ever and only converges inside that range, and the condition is part of the answer.
| 1. ⁵P₃ | 5 × 4 × 3 = 60. |
| 2. Committee of 3 from 10 | ¹⁰C₃ = 120. |
| 3. 4-digit PIN | 10⁴ = 10000. |
1 markSaying which it is and why, before any numbers.
1 markMultiplying independent stages rather than adding.
1 markStating the range of validity on an extended binomial expansion.
| They give | What it means |
|---|---|
| 10 (Q1) | Gave the combination when arrangements were asked for. |
| 720 (Q2) | Used a permutation; a committee has no order. |
| 5040 (Q3) | Forbade repeats, but a PIN may repeat digits. |
| 210 (Q3) | Ignored order entirely, and a PIN is very much ordered. |
| 40 (Q3) | Added the stages instead of multiplying. |
"Which formula do I use?" Neither, until you have answered the order question in words. The formula follows from the answer.
"Why divide by r factorial?" Because every unordered choice was counted once for each of its orderings. The figure shows the six copies being collapsed.
"Does the extended binomial ever stop?" No, and that is the difference. It is an approximation, improved by taking more terms, and only inside |x| < 1.
| What is happening | |
|---|---|
| 1 | Ask for the count. Fan them out. Name the 3!. |
| 2 | The order question, applied to five quick scenarios. |
| 3 | Product rule, with and without repetition. |
| 4 | The extended binomial, with the validity condition. |
| 5 | An approximation question using three terms. |
Do not teach nPr and nCr as two separate formulas. One is the other times r!, and the figure shows it.
Do not let an extended expansion be written without its range of x.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.