Selections / Combinations (Ex 10C)

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

Name

Class

Date

Learning intentions.

Decide selection (order doesn't matter) vs arrangement (order matters)
Use $^nC_r=\dfrac{n!}{r!\,(n-r)!}$ for teams and committees
Handle "some guaranteed" and "at least"; count selections of any size with $2^n$

Key results

$^nC_r=\dfrac{n!}{r!\,(n-r)!}$ and $^nC_r=\,^nC_{n-r}$. "Some guaranteed" → fix them, choose the rest. "At least one" → total $-$ none. Any size (incl. none) $=2^n$.

Warm-up — selection or arrangement?

1. Write S (selection) or A (arrangement), then give the value.

a A hand of $5$ cards from a deck

b The finishing order of $5$ runners

c A team of $3$ from $8$ players

d A $4$-letter password from $4$ distinct letters

2. Evaluate.

a $^8C_3$

b $^{10}C_4$

c $^{13}C_7$

Part A — Basic selections

3. A netball squad has $13$ players.

a How many teams of $7$?

b Explain why $^{13}C_7=\,^{13}C_6$.

4. An ice-cream parlour has $20$ flavours. How many three-scoop sundaes use $3$ different flavours (order doesn't matter)?

5. A first XI cricket team of $11$ is chosen from a squad of $15$. How many possible teams?

Part B — Some guaranteed & at least

6. A committee of $4$ is chosen from $10$ people, but one named person must be on it. How many committees?

Counting 10C — continued

Part B continued · Part C (any size) · Challenge

Name

Part B — Some guaranteed & at least (continued)

7. A team of $5$ is chosen from $5$ boys and $5$ girls.

a How many teams have exactly $2$ boys?

b How many have at least $1$ boy?

8. A group of $4$ is chosen from $6$ men and $4$ women, and must contain at least $2$ women. How many such groups? (Split into the cases: exactly 2, 3, 4 women.)

Part C — Selection of any size

9. A snack tray holds $6$ different items.

a How many selections of any size, including taking none?

b How many if you must take at least one item?

Challenge

10. A school must form a delegation by choosing $2$ of $5$ junior students, $3$ of $7$ middle students and $2$ of $6$ senior students.

a How many possible delegations? (Multiply the three selections.)

b Explain why you multiply the three counts rather than add them.