Foundations — Permutations — nPr, arrangements with restrictions

This page is a self-contained foundation for the topic Permutations — nPr, arrangements with restrictions (call it the topic below). That topic asks "how many ordered arrangements can I make?" and uses a handful of symbols and ideas without stopping to build them. Here we build every one of them from zero, in an order where each piece earns the next. Read top to bottom; skip nothing — no symbol is used before it is defined.

0. "Object" and "arrangement" — the raw picture

Picture five friends and five chalk marks on the floor labelled slot 1, slot 2, slot 3, slot 4, slot 5. An arrangement is one complete way of putting one friend on each mark.

Figure s01 — On the left, a pile of five distinct labelled objects (the things we will arrange). On the right, five empty ordered slots. An arrow shows one object walking from the pile into a slot: an arrangement is one full way to fill every slot, one object each.

Why the topic needs this: the whole subject is "how many different such pictures exist?" If you cannot see the slots, none of the symbols will mean anything.

1. The letter that counts your objects

Look at the left-hand pile in figure s01 — $n$ counts exactly those objects, before any are placed.

Why a letter and not a number? Because a formula must work for any pile size. Writing $n$ lets one rule cover 5 letters, 10 people, or a million cards.

2. The letter that counts your slots

In figure s01, if only the first three chalk marks were used and the last two stayed empty, that would be $r=3$ slots being filled out of $n=5$ objects.

3. Curly braces $\{\}$ and the word "distinct"

Why "distinct" matters here: the topic's formula $n{P}_r=\frac{n!}{(n-r)!}$ silently assumes no duplicates. If two objects were identical, swapping them would not make a new picture, and you'd over-count. (That repaired case is Permutations with Repetition.)

4. The multiplication principle — the engine

This is the whole Fundamental Counting Principle, and it is why we multiply choices instead of adding them.

Figure s02 — A branching tree for "2 shirts, 3 hats". From the start dot, two magenta branches (choice 1) reach the two shirts; from each shirt, three orange branches (choice 2) reach the three hats. Counting the endpoints gives $2\times 3=6$ complete paths — each path is one full outfit.

Why multiply, not add? Look at the tree in figure s02. Adding would answer "how many single decisions exist." Multiplying answers "how many complete paths from top to bottom exist" — and a full arrangement is a complete path. That is exactly what we want to count.

5. The factorial symbol $!$

Why the topic needs $!$: if you fill all $n$ slots, the choices are $n,n-1,n-2,\dots ,1$. Multiplying them (the multiplication principle!) is literally the definition of $n!$. So a factorial is just "the count of ways to line everyone up."

The special value $0!$

This is not a trick; it is the only value that keeps the formulas consistent (see Factorials and 0!). It makes $n{P}_n=n!$ and $n{P}_0=1$ come out right.

6. Subtraction inside the formula: $n-r$ and $n-r+1$

Figure s03 — Three snapshots of the same pile left to right. At slot 1 there are $n$ objects and $n$ choices; at slot 2 the pile has lost one, showing $n-1$; at slot 3 it shows $n-2$. Arrows between snapshots mark "one placement removes one object", so the choices count down.

Why $(n-r)!$ appears in the compact formula: after you have placed $r$ objects, $n-r$ are still in the pile, unused. The factorial $(n-r)!$ counts all the ways those leftovers could have been arranged — and dividing by it removes exactly the arrangements you don't care about. That is the single algebra move that turns the long product into $\frac{n!}{(n-r)!}$.

7. The notation $n{P}_r$ itself

$n{P}_r=\underset{r\text{ factors}}{\underbrace{n(n-1)\cdots (n-r+1)}}=\frac{n!}{(n-r)!}$

Why not just say a number? Because the question changes with $n$ and $r$. The symbol packages the whole slot-filling story into three characters.

8. Order vs no-order — the fork in the road

Think of $AB$ and $BA$: they are two different arrangements on a permutation list, but they collapse to one selection on a combination list. The link between them is $n{C}_r=\frac{n{P}_r}{r!}$ — you divide out the $r!$ internal orderings you no longer care about. This is exactly the leftover-factorial idea from §6, reused. (Details in Combinations — nCr.)

Prerequisite map

The diagram below shows how these foundations feed the topic. Read each arrow as "you need the left box before the right box makes sense." In words: objects and slots (§0) give birth to the counters $n$ (§1) and $r$ (§2); those two feed the multiplication principle (§4); the multiplication principle produces the shrinking pile (§6), which in turn defines factorials and $0!$ (§5); factorials plus the multiplication principle assemble into $n{P}_r$ (§7); and the order-matters question (§8) is the gate that decides you are doing $n{P}_r$ at all. If the picture ever fails to load, that sentence is the whole map.

The topic $n{P}_r$ sits at the mouth of all of these prerequisites.

Equipment checklist

Self-test: can you answer each before revealing? If any is shaky, reread that section.

Connections

• Fundamental Counting Principle — the multiplication engine of §4.
• Factorials and 0! — the symbol $!$ and why $0!=1$.
• Combinations — nCr — the "order ignored" fork from §8.
• Permutations with Repetition — what happens when objects are not distinct.
• Circular Permutations — arrangements in a ring instead of a line.
• Probability — Equally Likely Outcomes — where these counts become denominators.
• Permutations — nPr, arrangements with restrictions — the parent this page equips you for.