STA1008S: Statistics for Engineers
University of Cape Town
Study guide, definitions & Notes
Permutations, Combinations, Conditional Probability, and Independence
Permutations: arrangements of objects in order. Arranging people, numbers, digits, letters, and
colours.
NB* Order of objects matters.
n permutations
𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) … … (𝑛 − 𝑘)
Example:
If there are 5 chairs of different colours. List all the possible arrangements.
There are 5! = 5 × 4 × 3 × 2 × 1 = 120 distinct arrangement of chairs.
n permutations taken r at a time
It is the number of ways of choosing and arranging r objects out of n distinguishable objects.
𝑛! 𝑛(𝑛 − 1)(𝑛 − 2) × … . .× (𝑛 − 𝑟 − 1) × (𝑛 − 𝑟)(𝑛 − 𝑟 − 1) × … .× 3 × 2 × 1
(𝑛)𝑟 = =
(𝑛 − 𝑟)! (𝑛 − 𝑟)(𝑛 − 𝑟 − 1) × … .× 3 × 2 × 1
Example:
A photographer takes 3 pictures at a time. How many pictures will he take if they are 10 people?
There are 10 objects taken 3 at a time.
10! 10!
(10)3 = = = 720 𝑝𝑖𝑐𝑡𝑢𝑟𝑒𝑠
(10 − 3)! 7!
Combinations: grouping of objects.
NB* Order of objects does not matter.
n combinations taken r at a time:
The number of ways of choosing r elements out of n elements without regard to the arrangement of
the chosen elements.
𝑛 𝑛!
( )=
𝑟 (𝑛
𝑟! − 𝑟)!
, Example:
How many ways can a 9 men team be formed from 15 men?
There are 15 men taken 9 at a time where order does not matter.
15 15!
( )= = 5005 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑓𝑜𝑟𝑚𝑖𝑛𝑔 𝑎 9 𝑚𝑒𝑛 𝑡𝑒𝑎𝑚 𝑓𝑟𝑜𝑚 15 𝑚𝑒𝑛.
9 9! (15 − 9)!
Permutations with repetitions
The number of permutations of n types of objects taken r at a time, allowing repetition.
𝑛𝑟 = 𝑛 × 𝑛 × 𝑛 × … … × 𝑛
Example:
How many four-digit numbers can be made from the 10 digits from 0 to 9, if repetitions are
permitted?
We have 4 slots to fill. Repetition of each digit is allowed. Then 10 10 10 10
104 = 10 000 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑚𝑎𝑑𝑒
Combinations with repetitions
The number of selections of r objects allowing for repetition.
𝑛+𝑟−1
( )
𝑟
Example:
A supermarket sells 10 types of jam. You buy three tins. How many combinations are possible?
The number of combinations of 10 jams taken 3 at a time, allowing repetitions.
10 + 3 − 1 12 12!
( )=( )= = 220 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠
3 3 3! (12 − 3)!
Counting Rules
1.The number of distinguishable arrangements of n items, of which
𝑛 𝑖𝑡𝑒𝑚𝑠 𝑜𝑓 𝑤ℎ𝑖𝑐ℎ 𝑛1 𝑎𝑟𝑒 𝑜𝑓 𝑜𝑛𝑒 𝑘𝑖𝑛𝑑 𝑎𝑛𝑑 𝑛2 = 𝑛 − 𝑛2 𝑎𝑟𝑒 𝑜𝑓 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑘𝑖𝑛𝑑.
𝑛! 𝑛 𝑛
=( )=( )
𝑛1 ! 𝑛2 ! 𝑛1 𝑛2
𝑛1 𝑎𝑛𝑑 𝑛2 𝑖𝑡𝑒𝑚𝑠 𝑎𝑟𝑒 𝑖𝑛𝑑𝑖𝑠𝑡𝑖𝑛𝑔𝑢𝑖𝑠ℎ𝑎𝑏𝑙𝑒 𝑓𝑟𝑜𝑚 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟
University of Cape Town
Study guide, definitions & Notes
Permutations, Combinations, Conditional Probability, and Independence
Permutations: arrangements of objects in order. Arranging people, numbers, digits, letters, and
colours.
NB* Order of objects matters.
n permutations
𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) … … (𝑛 − 𝑘)
Example:
If there are 5 chairs of different colours. List all the possible arrangements.
There are 5! = 5 × 4 × 3 × 2 × 1 = 120 distinct arrangement of chairs.
n permutations taken r at a time
It is the number of ways of choosing and arranging r objects out of n distinguishable objects.
𝑛! 𝑛(𝑛 − 1)(𝑛 − 2) × … . .× (𝑛 − 𝑟 − 1) × (𝑛 − 𝑟)(𝑛 − 𝑟 − 1) × … .× 3 × 2 × 1
(𝑛)𝑟 = =
(𝑛 − 𝑟)! (𝑛 − 𝑟)(𝑛 − 𝑟 − 1) × … .× 3 × 2 × 1
Example:
A photographer takes 3 pictures at a time. How many pictures will he take if they are 10 people?
There are 10 objects taken 3 at a time.
10! 10!
(10)3 = = = 720 𝑝𝑖𝑐𝑡𝑢𝑟𝑒𝑠
(10 − 3)! 7!
Combinations: grouping of objects.
NB* Order of objects does not matter.
n combinations taken r at a time:
The number of ways of choosing r elements out of n elements without regard to the arrangement of
the chosen elements.
𝑛 𝑛!
( )=
𝑟 (𝑛
𝑟! − 𝑟)!
, Example:
How many ways can a 9 men team be formed from 15 men?
There are 15 men taken 9 at a time where order does not matter.
15 15!
( )= = 5005 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑓𝑜𝑟𝑚𝑖𝑛𝑔 𝑎 9 𝑚𝑒𝑛 𝑡𝑒𝑎𝑚 𝑓𝑟𝑜𝑚 15 𝑚𝑒𝑛.
9 9! (15 − 9)!
Permutations with repetitions
The number of permutations of n types of objects taken r at a time, allowing repetition.
𝑛𝑟 = 𝑛 × 𝑛 × 𝑛 × … … × 𝑛
Example:
How many four-digit numbers can be made from the 10 digits from 0 to 9, if repetitions are
permitted?
We have 4 slots to fill. Repetition of each digit is allowed. Then 10 10 10 10
104 = 10 000 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑚𝑎𝑑𝑒
Combinations with repetitions
The number of selections of r objects allowing for repetition.
𝑛+𝑟−1
( )
𝑟
Example:
A supermarket sells 10 types of jam. You buy three tins. How many combinations are possible?
The number of combinations of 10 jams taken 3 at a time, allowing repetitions.
10 + 3 − 1 12 12!
( )=( )= = 220 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠
3 3 3! (12 − 3)!
Counting Rules
1.The number of distinguishable arrangements of n items, of which
𝑛 𝑖𝑡𝑒𝑚𝑠 𝑜𝑓 𝑤ℎ𝑖𝑐ℎ 𝑛1 𝑎𝑟𝑒 𝑜𝑓 𝑜𝑛𝑒 𝑘𝑖𝑛𝑑 𝑎𝑛𝑑 𝑛2 = 𝑛 − 𝑛2 𝑎𝑟𝑒 𝑜𝑓 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑘𝑖𝑛𝑑.
𝑛! 𝑛 𝑛
=( )=( )
𝑛1 ! 𝑛2 ! 𝑛1 𝑛2
𝑛1 𝑎𝑛𝑑 𝑛2 𝑖𝑡𝑒𝑚𝑠 𝑎𝑟𝑒 𝑖𝑛𝑑𝑖𝑠𝑡𝑖𝑛𝑔𝑢𝑖𝑠ℎ𝑎𝑏𝑙𝑒 𝑓𝑟𝑜𝑚 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟
