Combinatorics Through Guided Discovery, 2004a

24 1. What is Combinatorics?
(vii) How many terms of the form x n−i y i will you have?
(c) Explain how you have just proved your conjecture from Problem 53.
The theorem you have proved is called the Binomial Theorem.
Problem 55. What is ∑ 10
i=1 (10 i
)3 i ? (h)
Problem 56. What is ( n 0 ) − (n 1 ) + (n 2 ) −···±(n )
n
if n is an integer bigger than
0? (h)
Problem 57. Explain why
m∑ ( m
i
i=0
)( ) n
=
k − i
( ) m + n
.
k
Find two different explanations. (h)
⇒
Problem 58. From the symmetry of the binomial coefficients, it is not too
hard to see that when n is an odd number, the number of subsets of
{1, 2,...,n} of odd size equals the number of subsets of {1, 2,...,n} of even
size. Is it true that when n is even the number of subsets of {1, 2,...,n} of
even size equals the number of subsets of odd size? Why or why not? (h)
⇒
Problem 59.
calculus.) (h)
What is ∑ n
i=0 i(n )?
i
(Hint: think about how you might use
Notice how the proof you gave of the binomial theorem was a counting argument.
It is interesting that an apparently algebraic theorem that tells us how to
expand a power of a binomial is proved by an argument that amounts to counting
the individual terms of the expansion. Part of the reason that combinatorial
mathematics turns out to be so useful is that counting arguments often underlie
important results of algebra. As the algebra becomes more sophisticated, so do the
families of objects we have to count, but nonetheless we can develop a great deal
of algebra on the basis of counting.