Combinatorics Through Guided Discovery, 2004a

2.1. Some Examples of Mathematical Induction 33
You will find some explicit examples of the use of the strong principle of mathematical
induction in Appendix B and will find some uses for it in this chapter.
2.1.2 Binomial Coefficients and the Binomial Theorem
• Problem 72. When we studied the Pascal Equation and subsets in Chapter 1,
it may have appeared that there is no connection between the Pascal relation
( n k ) = (n−1 k−1 ) + (n−1 k ) and the formula (n k ) = n! . Of course you probably
k!(n−k)!
realize you can prove the Pascal relation by substituting the values the
formula gives you into the right-hand side of the equation and simplifying
to give you the left hand side. In fact, from the Pascal Relation and the facts
that ( n 0 ) =1and (n n ) =1, you can actually prove the formula for (n k ) by
induction on n. Do so. (h)
⇒ Problem 73. Use the fact that (x + y) n = (x + y)(x + y) n−1 to give an
inductive proof of the binomial theorem. (h)
Problem 74. Suppose that f is a function defined on the nonnegative integers
such that f (0) = 3 and f (n) =2f (n − 1). Find a formula for f (n) and
prove your formula is correct.
Problem 75. Prove the conjecture in Part 13.b for an arbitrary positive integer
m without appealing to the general product principle. (h)
2.1.3 Inductive definition
You may have seen n! described by the two equations 0!=1and n! =n(n − 1)!
for n > 0. By the principle of mathematical induction we know that this pair of
equations defines n! for all nonnegative numbers n. For this reason we call such
a definition an inductive definition. An inductive definition is sometimes called
a recursive definition. Often we can get very easy proofs of useful facts by using
inductive definitions.
⇒
Problem 76. An inductive definition of a n for nonnegative n is given by
a 0 =1and a n = aa n−1 . (Notice the similarity to the inductive definition of
n!.) We remarked above that inductive definitions often give us easy proofs
of useful facts. Here we apply this inductive definition to prove two useful
facts about exponents that you have been using almost since you learned
the meaning of exponents.