1. Counting techniques
In general to find out the number of outcomes in a multistage event you multiply the number of outcomes from the first event by the number of outcomes in the second event and so on.
Today’s questions assume that there is NO replacement.
This means that once an item has been selected, that item can’t be selected again in that.
Examples may include:
Drawing two numbers out of a hat to form a two digit number
People are standing for positions such as president & secretary
Putting books on a book shelf
Horses finishing a race
In all of these examples ORDER is important.

2. Factorial notation (n!)
The number of ways that n items can be arranged if ALL of the items are used is given by n!.
For example, how many orders can I read 3 books?
3 × 2 × 1 = 6
This can be done using the x! button on your calculator. Type in:
3 shift x! (on the top left of your calculator).
Example 1
The Melbourne cup has 24 horses. How many ways can they finish the race?
24! =
6·20448 × 1023 (6 sig fig)

3. Ordered selections (Permutations nPr)
The number of ways that n items can be arranged if SOME of the items are used is given by
n × (n − 1) × (n − 2) × (n − 3) ….
For example, how many orders can I read 3 books from a selection of 10?
10 × 9 × 8 = 720
This can be done using the nPr button on your calculator. Type in:
10 shift nPr 3 (on the top left of your calculator).
Example 2
The Melbourne cup has 24 horses and a trifecta is picking the first 3 horses in order. How trifectas are there?
24 × 23 × 22 =
12 144
or 24P3 =
12 144

4. Example 3 Example 4 9 greyhounds are in a race.
How many ways can the dogs finish the race?
How many different trifectas are there?
9! =
9 × 8 × 7 =
or 9P3 =
504
504
Example 4
4 girls and 3 boys line up in a bank.
How many ways can the people line up?
How many ways can all the girls line up in front of all the boys?
How many ways can all the girls line up together?
How many ways can girls and boys alternate?
7! =
4! × 3! =
4! × 4! =
5040
Just line up 7 people in a definite order
144
Line up 4 girls in a definite order, then 3 boys
576
4 girls together, but they can mix in with the 3 boys
144
GBGBGBG, treat them as 2 separate groups.

5. Today’s work Exercise 7C Page 215 Q1, 3, 5, 7, 9, 11 Exercise 7D