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A basketball coach has
20 players available.

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Out of them, he needs to choose
five for the starting

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lineup, and seven who would
be sitting on the bench.

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In how many ways can the coach
choose these 5 plus 7 players?

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It is certainly a huge number,
but what exactly is it?

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In this lecture, we will
learn how to answer

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questions of this kind.

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More abstractly, we will develop
methods for counting

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the number of elements of a
given set which is described

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in some implicit way.

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Now, why do we care?

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The reason is that in many
models, the calculation of

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probabilities reduces
to counting.

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Counting the number of elements
of various sets.

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Suppose that we have a
probability model in which the

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sample space, Omega, is finite,
and consists of n

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equally likely elements.

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So each element has
probability 1/n.

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Suppose now that we're
interested in the probability

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of a certain set, A, which
has k elements.

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Since each one of the elements
of A has probability 1/n, and

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since A has k distinct
elements, then by the

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additivity axiom, the
probability of A is equal to k

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times 1 over n.

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Therefore to find the
probability of A, all we have

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to do is to count the number of
elements of Omega and the

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number of elements of A,
and so determine the

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numbers k and n.

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Of course, if a set is described
explicitly through a

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list of its elements, then
counting is trivial.

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But when a set is given
through some abstract

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description, as in our
basketball team example,

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counting can be a challenge.

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In this lecture, we will start
with a powerful tool, the

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basic counting principle, which
allows us to break a

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counting problem into
a sequence of

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simpler counting problems.

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We will then count permutations,
subsets,

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combinations, and partitions.

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We will see shortly what all
of these terms mean.

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In the process we will solve a
number of example problems,

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and we will also derive the
formula for the binomial

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probabilities, the probabilities
that describe

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the number of heads
in a sequence of

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independent coin tosses.

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So, let us get started.