400 Statistics Lecture 3 Methods of Enumeration
Multiplication Principle: Suppose that an experiment E1 has n1 outcomes and an experiment E2 has n2 possible outcomes. The composite experiment E1E2 has n1*n2 outcomes. Example: There are 5 seats in a classroom and 5 students registered for a class. How many seating arrangements do we have?
Definition 1.3-1 Each of the n! arrangements of n different objects is called a permutation of the n objects.
Example: There are only 3 seats in a classroom and 5 students registered for a class. How many seating arrangements do we have?
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Definition 1.3-2 Each of the nPr arrangements is called a permutation of the n objects taken r at a time.
n
Pr =
n! (n - r )!
Example: The number of ways of selecting a president, a vice president in a club which has 10 members.
Definition 1.3-3 If r objects are selected from a set of n objects, and the order of selection is noted, the selected set of r objects is called an ordered sample of size r. Definition 1.3-4 Sampling with replacement: one object is selected and put back before the next is one selected. Definition 1.3-5 Sampling without replacement: one object is selected and not put back before the next one is selected.
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Example: The number of all possible four-letter words selected from 26 letters is
If the four letters are different, then how many possible words?
Example: The number of ways of selecting two presidents in a club which has 10 members.
Definition 1.3-6 Each of the nCr unordered subsets is called a combination of n objects taken r at a time where
n
Cr =
n! r!( n - r )!
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The number nCr are also called binomial coefficients
( a + b) n =
Example: Draw 5 cards f...