Combinatorial Probabilities Cheat Sheet

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Math 408, Actuarial Statistics I
A.J. Hildebrand
Combinatorial Probabilities
Key concepts
Permutation: arrangement in some order.
Ordered versus unordered samples: In ordered samples, the order of the elements
in the sample matters; e.g., digits in a phone number, or the letters in a word. In
unordered samples the order of the elements is irrelevant; e.g., elements in a subset, or
lottery numbers.
Samples with replacement versus samples without replacement: In the first
case, repetition of the same element is allowed (e.g., numbers in a license plate); in the
second, repetition not allowed (as in a lottery drawing—once a number has been drawn,
it cannot be drawn again).
Formulas
Number of permutations of
objects: !
Number of ordered samples of size , with replacement, from
objects:
Number of ordered samples of size , without replacement, from
objects:
!
(
1)
(
+ 1) =
=
(
)!
Number of unordered samples of size , without replacement, from a set of
objects
(= number of subsets of size
from a set of
elements) (combinations):
!
(
1)
(
+ 1)
=
=
=
!
!(
)!
!
(See back of page for properties of these binomial coefficients.)
Number of subsets of a set of
elements: 2
1

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