Count without counting twice

Multiply the number of choices at successive stages. Add mutually exclusive cases. If cases overlap, subtract the overlap once using inclusion–exclusion.

Order matters for a permutation; a combination counts a selection. With repeated objects, divide by the factorial of each repetition count. State restrictions before calculating.

\displaystyle {}^nP_r=\frac{n!}{(n-r)!},\quad {n\choose r}=\frac{n!}{r!(n-r)!}

WORKED EXAMPLE

Choose a captain and vice-captain from seven people.

Choose the captain in 7 ways.
Choose the vice-captain from the remaining 6.
There are 7 × 6 = 42 ordered selections.

Assumed knowledge

List outcomes systematically, use simple tree diagrams, and multiply the numbers of choices at successive stages. Be comfortable with basic probability and factorial notation.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Count without counting twice

Choose a captain and vice-captain from seven people.

Put the idea to work.