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Mathematics and Statistics. We introduce these concepts here because this is how the factorials
make their way into the Binomial Theorem, as our next deﬁnition indicates.
Definition 9.5. Binomial Coeﬃcients: Given two whole numbers n and j with n ≥ j , the
(read, n choose j ) is the whole number given by
j !(n − j )!
The name ‘binomial coeﬃcient’ will be justiﬁed shortly. For now, we can physically interpret
j as the number of ways to select j items from n items where the order of the items selected is
unimportant. For example, suppose you won two free tickets to a special screening of the latest
Hollywood blockbuster and have ﬁve good friends each of whom would love to accompany you to
the movies. There are 5 ways to choose who goes with you. Applying Deﬁnition 9.5, we get
2 = 5!
2!(5 − 2)!
2 So there are 10 diﬀerent ways to distribute those two tickets among ﬁve friends. (Some will see it
as 10 ways t...
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