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 nobjects: n!
•Number of ordered samples of size r,with replacement, from nobjects: nr
•Number of ordered samples of size r,without replacement, from nobjects:
n(n−1) · · · (n−r+ 1) = n!
(n−r)! =nPr.
•Number of unordered samples of size r,without replacement, from a set of nobjects
(= number of subsets of size rfrom a set of nelements) (combinations):
n
r=nPr
r!=n!
r!(n−r)! =n(n−1) . . . (n−r+ 1)
r!.
(See back of page for properties of these binomial coefficients.)
•Number of subsets of a set of nelements: 2n
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