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Foundations of Mathematics
Summer Program , Columbia University
July 1999 -- Handout #1 for Week #3
Roger B. Blumberg

Theorem 1: (The Fundamental Theorem of Counting): Suppose the set A1 contains n1 members, that the set A2 contains n2 members, the set A3 contains n3 members, ......., and the set A(m) contains n(M) members. Then the number of ways to choose one member from each of the m sets is equal to:

n1 x n2 x n3 x ........... x n(M)
Theorem 2: The number of permutations of n members of a set is equal to n!.
Theorem 3: The number of permutations of n members of a set arranged in r places is equal to:

P(n,r) = n!/(n-r)!
Theorem 4: The number of combinations of n members of a set, chosen r at a time, is equal to:

C(n,r) = n!/[(n-r)! * r!]
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