Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
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24 1. What is <strong>Combinatorics</strong>?<br />
(vii) How many terms of the form x n−i y i will you have?<br />
(c) Explain how you have just proved your conjecture from Problem 53.<br />
The theorem you have proved is called the Binomial Theorem.<br />
Problem 55. What is ∑ 10<br />
i=1 (10 i<br />
)3 i ? (h)<br />
Problem 56. What is ( n 0 ) − (n 1 ) + (n 2 ) −···±(n )<br />
n<br />
if n is an integer bigger than<br />
0? (h)<br />
Problem 57. Explain why<br />
m∑ ( m<br />
i<br />
i=0<br />
)( ) n<br />
=<br />
k − i<br />
( ) m + n<br />
.<br />
k<br />
Find two different explanations. (h)<br />
⇒<br />
Problem 58. From the symmetry of the binomial coefficients, it is not too<br />
hard to see that when n is an odd number, the number of subsets of<br />
{1, 2,...,n} of odd size equals the number of subsets of {1, 2,...,n} of even<br />
size. Is it true that when n is even the number of subsets of {1, 2,...,n} of<br />
even size equals the number of subsets of odd size? Why or why not? (h)<br />
⇒<br />
Problem 59.<br />
calculus.) (h)<br />
What is ∑ n<br />
i=0 i(n )?<br />
i<br />
(Hint: think about how you might use<br />
Notice how the proof you gave of the binomial theorem was a counting argument.<br />
It is interesting that an apparently algebraic theorem that tells us how to<br />
expand a power of a binomial is proved by an argument that amounts to counting<br />
the individual terms of the expansion. Part of the reason that combinatorial<br />
mathematics turns out to be so useful is that counting arguments often underlie<br />
important results of algebra. As the algebra becomes more sophisticated, so do the<br />
families of objects we have to count, but nonetheless we can develop a great deal<br />
of algebra on the basis of counting.