Arrangements / Permutations (Ex 10B)

Year 11 Mathematical Methods · Cambridge Methods 1&2, Counting Methods · Mr Wong

Name

Class

Date

Learning intentions.

Use factorial notation $n!$ to arrange $n$ distinct objects in a row
Use $^nP_r=\dfrac{n!}{(n-r)!}$ to arrange $r$ of $n$ objects
Handle restrictions: items together, items apart, even / end-position conditions

Key results

$n!=n(n-1)\cdots2\cdot1$ and $0!=1$. $^nP_r=\dfrac{n!}{(n-r)!}$. "Together" → glue into a block then $\times k!$; "not together" → total $-$ together.

Warm-up — factorials and nPr

1. Evaluate.

a $4!$

b $6!$

c $7!$

d $^7P_2$

e $^8P_3$

f $^{12}P_4$

2. Four people line up at a counter. Fill the boxes, then state the number of orders.

Part A — Arranging in a row

3. Seven different books are placed on a shelf.

a How many arrangements?

b How many if only $3$ of the $7$ are displayed in a row?

4. A relay coach must choose an ordered team for legs $1,2,3,4$ from $9$ athletes. In how many ways? (Use $^nP_r$.)

5. How many arrangements are there of the $6$ distinct letters of the word NUMBER?

Part B — Restrictions: even & end positions

6. Each of the digits $1,2,3,4,5$ is used exactly once to form a five-digit number.

a How many such numbers are even?

b How many are greater than $30\,000$?

Counting 10B — continued

Part C (together / apart) · Challenge

Name

Part C — Items together or apart

7. Five students $A,B,C,D,E$ sit in a row.

a Total arrangements (no restriction)

b Arrangements with $A$ and $B$ together

c Arrangements with $A$ and $B$ not together

8. Eight people sit in a row, but two particular friends must sit together. How many arrangements? (Glue the pair into a block.)

9. Eight boys and two girls sit in a row.

a Arrangements with the two girls together

b Arrangements with the two girls not together

Challenge

10. Eight students are to sit in a row. Two particular students, $A$ and $B$, must not sit next to each other, and a third student, $C$, must sit in one of the two end seats.

a How many seating arrangements satisfy both conditions?

b Explain your strategy and how you avoided over-counting.