**Unformatted text preview: **j !(n − j + 1)! = n! (j + (n − j + 1))
j !(n − j + 1)! = (n + 1)n!
j !(n + 1 − j ))! = (n + 1)!
j !((n + 1) − j ))! n+1
j
We are now in position to state and prove the Binomial Theorem where we see that binomial
coeﬃcients are just that - coeﬃcients in the binomial expansion.
Theorem 9.4. Binomial Theorem: For nonzero real numbers a and b and natural numbers n,
= n
n (a + b) =
j =0 n n−j j
ab
j To get a feel of what this theorem is saying and how it really isn’t as hard to remember as it may
ﬁrst appear, let’s consider the speciﬁc case of n = 4. According to the theorem, we have
4 (a + b)4 =
j =0 4 4−j j
ab
j = 4 4−0 0
4 4−1 1
4 4−2 2
4 4−3 3
4 4−4 4
a b+
a b+
a b+
a b+
ab
0
1
2
3
4 = 44
43
4 22
4
44
a+
a b+
ab +
ab3 +
b
0
1
2
3
4 9.4 The Binomial Theorem 585 We forgo the simpliﬁcation of the coeﬃcients in order to note the pattern in the expansion. First
note that in each term, the total of the exponents is 4 which matched the exponent of the binomial
(a + b)4 . The exponent on a begins at 4 and decreases by one as we move from one term to t...

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