Counting techniques In general to find out the

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.

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)

Ordered selections (Permutations n. Pr) 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 n. Pr button on your calculator. Type in: 10 shift n. Pr 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 24 P 3 = 12 144

Example 3 9 greyhounds are in a race. a) How many ways can the dogs finish the race? b) How many different trifectas are there? a) 9! = 362 880 b) 9 × 8 × 7 = 504 or 9 P 3 = 504 Example 4 4 girls and 3 boys line up in a bank. a) How many ways can the people line up? b) How many ways can all the girls line up in front of all the boys? c) How many ways can all the girls line up together? d) How many ways can girls and boys alternate? a) 7! = 5040 Just line up 7 people in a definite order b) 4! × 3! = 144 Line up 4 girls in a definite order, then 3 boys c) 4! × 4! = 576 4 girls together, but they can mix in with the 3 boys d) 4! × 3! = 144 GBGBGBG, treat them as 2 separate groups.

Today’s work Exercise 7 C Page 215 Q 1, 3, 5, 7, 9, 11 Exercise 7 D Page 216 Q 1, 3, 5, 7