Permutations and Combinations
Ordered Samples With Replacement
Situation: A box has n items numbered 1 , . . . , n . Select k items with replacement. (A number can be drawn twice).
Sample Space:
Ω = { ( x
1
, . . . , x k
) : x i
∈ { 1 , . . . , n }}
= { x
1 x
2
...x
k
: x i
∈ { 1 , . . . , n }}
What is | Ω | ?
Break the complex action into a series k single draws. ( x i is outcome on draw i ).
– n possibilities for x
1
– n possibilities for x
2
...
– n possibilities for x k
.
Multiplication principle → | Ω | = n k .
Example: Octal Numbers
A five-digit octal number is a 5-digit number consisting of the digits 0 , . . . , 7.
1. How many 5-digit octal numbers are there?
2. What is the probability that a randomly chosen 5-digit number is an octal number?
Example: Coin Toss
Toss a coin 23 times.
1. How many sequences of H 0 s and T 0 s are there?
Ordered Samples Without Replacement
Situation: A box has n items numbered 1 , . . . , n . Select k ≤ n items without replacement.
(A number is drawn at most once). Keep track of the sequence of selections.
Sample Space:
Ω = { ( x
1
, . . . , x k
) : x i
∈ { 1 , . . . , n } , x i
= x j
}
= { x
1 x
2
...x
k
: x i
∈ { 1 , . . . , n } x i
= x j
}
What is | Ω | ?
Break the complex action into a series k single draws. ( x i is outcome on draw i ).

– n possibilities for x
1
– n − 1 possibilities for x
2
...
– n − ( k − 1) possibilities for x k
.
Multiplication principle implies | Ω | = n ( n − 1)( n − 1) · · · ( n − ( k − 1)) = n !
( n − k )!
.
| Ω | has a name...
Definition:
Permutation - An ordering of k distinct objects chosen from n distinct objects.
Definition:
Permutation Number : P ( n, k ) - The number of permutations of k objects from n .
Theorem:
P ( n, k ) = n !
( n − k )!
Example: Pizza Toppings
A survey question lists seven pizza toppings and aks you to rank your favorite 3.
1. How many possible answers to the survey question are there?
2. One of the choices is “olives”. If all possible rankings are equally likely, what is the probability that a randomly selected survey has “olives” in the top 3?
Unordered Samples Without Replacement
Situation: A box has n items numbered 1 , . . . , n . Select k ≤ n items without replacement.
(A number is drawn at most once). Keep track of the set of numers selected in the end.
(Order does not matter).
Sample Space:
Ω = {{ x
1
, . . . , x k
} : x i
∈ { 1 , . . . , n } , x i
= x j
}
What is | Ω | ?
| Ω | = # collections of k distinct objects chosen from n.
P ( n, k ) can be broken into two steps:
– Select a collection of k from n .

– Order the k objects.
Multiplication Principle implies
P ( n, k ) = (# of collections of k from n ) × (# ways to order the k )
= | Ω | × P ( k, k ) .
| Ω | =
P ( n, k )
P ( k, k )
= n !
( n − k )!
k !
| Ω | has a name...
Definition:
Combination - A subset of n distinct objects which has k distinct objects.
Theorem. The number of combinations of k objects chosen from n is
C ( n, k ) = n k n !
=
( n − k )!
k !
Example: Powerball (without the powerball)
You pick 5 numbers from { 1 , . . . , 49 } without replacement. The lottery does the same. The order in which the numbers are picked is irrelevant. You win if you and the lottery pick at least three of the same numbers.
1. What is the probability that all 5 match?
2. What is the probability that you win?
Example: Coin Toss
Toss a coin 23 times.
1. How many ways are there to get 17 heads?
2. If the coin is fair, what is the probability of getting 17 heads?