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Material Type: Quiz; Class: Basic Discrete Mathematics; Subject: Mathematics; University: University of Illinois - Urbana-Champaign; Term: Fall 2008;
Typology: Quizzes
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Math 213, Section B1, Quiz 4 (Solution); Friday, February 8, 2008
Prove that for every integer n โฅ 1 we have
โ^ n
j=
n
j
j = 6
n โ 1.
Solution.
By the Binomial Theorem for every x, y โ R we have
(x + y)
โ^ n
j=
n
j
x
nโj y
j .
Substituting x = 1, y = 5 in the above formula, we get:
n = (1 + 5)
โ^ n
j=
n
j
nโj 5
โ^ n
j=
n
j
n
0
0
โ^ n
j=
n
j
j = 1 +
โ^ n
j=
n
j
j .
Therefore โn
j=
n
j
j = 6
n โ 1 ,
as required.
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