Mixed arrangements
There are five different ways to systematically determine the number
of outcomes in a multi stage event:
* Write a list of all possibilities and count them.
* Draw a tree diagram to help you systematically list all possibilities
* Multiply the number of possibilities at each stage of the
multistage event.
* Use ‘combinations’ and the nCr button when choosing part of
the group and order is not important.
* Use ‘permutations’ and the nPr button when choosing part of the
group and order is important.
Mixed arrangements
You must be able to distinguish between permutations and
combinations.
Permutations are ordered arrangements. Here the order within the
group is imPortant.
Here ABC, ACB and CBA are considered different arrangements.
Examples include:
•Picking a trifecta in a 10 horse race.
•Picking president and secretary from a group of 6 people.
Combinations are unordered selections. Here the order within the
group is not important (or who Cares).
Here ABC, ACB and CBA are considered the same.
Examples include:
•Picking 3 horses in a 10 horse race.
•Picking a committee of 2 from a group of 6 people.
Example 1
In the “pools” you need to pick 6 numbers from 38.
In the strike version of the game you need to pick the first 4 balls in the
order in which they come out of the container.
a) How many different ways are there of selecting the 6 balls?
b) How many different strike games could you play?
c) In a system 8 you pick from only 8 numbers. How many different
combinations of 6 numbers is this?
d) If Sarah plays a system 8, what is the probability that she wins?
a)
b)
38C =
6
38P =
4
2 760 681
1 771 560
c) 8C6 = 28
d) P(win with system 8) =
28
2 760 681
4

This is about 1 in 98 600
394 383
Example 2
In poker you are dealt a hand of 5 cards.
a) How many different hands are possible?
b) How many ways can you pick 3 kings from the 4 in the deck?
c) How many ways can you pick 2 jacks from the 4 in the deck?
d) What is the probability of being dealt 3 kings and 2 jacks?
e) How many ways can you pick 5 cards from the same suit?
f) A flush is 5 cards from the same suit. What is the probability of
being dealt a flush in hearts?
g) What is the probability of being dealt a flush?
a) 52C5 = 2 598 960
e) 13C5 = 1287
1287
b) 4C3 = 4
f) P(Flush H) =
4
2 598 960
c) C2 = 6
d) Ways of selecting 3K & 2J = 4 × 6 =24 g) P(Flush) = 4  1287
1
2 598 960
24

Times by 4 as
P(3K, 2J) =
33
108
290
2 598 960
there are 4 suits 
16 660
Example 3
Kathini has 4 english books and 2 maths books.
a) How many ways can the books be arranged on the shelf ?
b) How many ways are the 4 english books together?
c) What is the probability the maths books are together?
a) 6! = 720
b) 4! × 3! =144 4 english books, now there are 3 groups of books
1 english and 2 maths
c) 2! × 5! =240 How many ways can the maths books be together?
2 maths books, now there are 5 groups of books
4 english and 1 maths
240
P(maths together) =
720
1

3
Today’s work
Permutations and Combinations Worksheets