Year 11 Mathematical Methods — Arrangements (Permutations)

Cambridge Methods 1&2 — Counting Methods (10B)  •  Sun 14 June 2026
📚 Also for this topic: 📄 Printable worksheet ✅ Solutions (answer key) → Next: 10C Selections

Today's lesson

An arrangement (or permutation) is an ordering of objects in a row — order matters. Today we build the multiplication principle into a neat shortcut: factorial notation $n!$, and the rule $^nP_r$ for arranging only some of the objects.

Learning intentions

Part 1 — Factorials: arranging $n$ in a row

How many ways can $4$ people line up? Fill four boxes: $4$ choices for the first seat, then $3$ left, then $2$, then $1$.

4
1st
×
3
2nd
×
2
3rd
×
1
4th
=
24
orders
Factorial. $n! = n\times(n-1)\times(n-2)\times\cdots\times2\times1$. The number of ways to arrange $n$ distinct objects in a row is $n!$. By definition $0!=1$.

For example $5!=120$, $\;6!=720$, $\;7!=5040$. Factorials grow very fast — $9!=362\,880$.

📺 Walkthrough: filling the boxes to arrange $4$ objects in a row — why the choices shrink $4,3,2,1$ and give $4!=24$.

Part 2 — $^nP_r$: arranging some of the objects

Often we only fill some positions. From $8$ swimmers, in how many ways can the first $3$ places be filled? Boxes: $8\times7\times6=336$. We can write this with factorials:

Permutations. The number of ways to arrange $r$ of $n$ distinct objects in order is $$^nP_r=\frac{n!}{(n-r)!}=n\times(n-1)\times\cdots\times(n-r+1)\quad(r\text{ factors}).$$

Check: $^8P_3=\dfrac{8!}{5!}=8\times7\times6=336.$   Note $^nP_n=n!$ (arranging all of them) since $0!=1$.

On your CAS. Most calculators have a built-in permutation function written nPr (look in the probability/maths menu). Enter nPr(8, 3) to get $336$. Use it to check your by-hand box working.
EXAMPLE 1 — factorials and nPr
(a) In how many orders can $10$ photos be hung in a row? (b) From $10$ photos, how many ways to choose and hang just $3$ in a row?
  1. (a) All $10$ in a row: $10! = 3\,628\,800$.
  2. (b) Arrange $3$ of $10$: $^{10}P_3=\dfrac{10!}{7!}=10\times9\times8=\boxed{720}$.

Now you try: From $7$ runners, in how many orders can $1$st, $2$nd, $3$rd, $4$th be awarded?   Answer: $^7P_4=7\times6\times5\times4=840$.

Part 3 — Arrangements with restrictions

Restrictions are handled by dealing with the restricted positions first, then filling the rest. Two classics: items that must be together, and items that must be apart.

EXAMPLE 2 — even numbers (end-digit restriction)
Using each of the digits $1,2,3,4,5$ exactly once, how many five-digit numbers are even?
  1. "Even" restricts the last digit — it must be $2$ or $4$, so $2$ choices. Deal with it first.
  2. The remaining $4$ digits fill the other $4$ places in $4!=24$ ways.
  3. Multiply: $2\times4! = 2\times24 = \boxed{48}$ even numbers.

Now you try: How many of those five-digit numbers are greater than $30\,000$?   Answer: the first digit must be $3,4$ or $5$ ($3$ choices), so $3\times4!=72$.

EXAMPLE 3 — items together / apart
Five students $A,B,C,D,E$ sit in a row. (a) In how many arrangements do $A$ and $B$ sit together? (b) In how many do they not sit together?
  1. (a) Together — "glue" them. Treat $AB$ as a single block, so we arrange $4$ items: $4!=24$. The block can be $AB$ or $BA$, i.e. $2!=2$ internal orders. Total $=4!\times2!=\boxed{48}$.
  2. (b) Not together — use the complement. All arrangements $=5!=120$. Subtract the "together" ones: $120-48=\boxed{72}$.

Key trick: "together" → glue into a block then multiply by the block's internal $k!$. "Not together" → total minus together.

Practice 3.1 — evaluate (use boxes, factorials or your CAS).

  1. $6!$
  2. $^7P_4$
  3. $^{12}P_4$
  4. Arrangements of the $6$ distinct letters of NUMBER
  5. Five-digit numbers from $1,2,3,4,5$ (no repeats) that end in $5$
  6. Six people in a row with two particular people together
a) $720$    b) $^7P_4=840$    c) $^{12}P_4=11\,880$    d) $6!=720$    e) fix $5$ at the end, $4!=24$    f) $5!\times2!=240$

Part 4 — Quick quiz (5 min)

Pick the correct answer for each, then click Mark.

Q1. The value of $5!$ is:

Q2. The number of ways to arrange $3$ of $8$ distinct objects in a row is:

Q3. Using $1,2,3,4,5$ once each, how many five-digit numbers are even?

Q4. Five people in a row; two particular people sit together. Number of arrangements?

Q5. $^nP_n$ equals:

Working program — Cambridge — Counting Methods Ex 10B

After the quiz, open Cambridge Methods 1&2 (Counting Methods, Chapter 10) and complete the set work:

Set workExtension
Ex 10B — Q1, 2, 3, 5, 8, 9, 13, 15Harder restriction questions and the chapter-review permutations.

For restriction questions, always deal with the restricted position(s) first.

Exit ticket — write in your book

Before you pack up, write one sentence each:
  1. What does $n!$ count, and what is $0!$?
  2. Write the formula for $^nP_r$ and explain what $r$ is.
  3. How do you count arrangements where two items must sit together?

— Mr Wong