Concrete mathematics : a foundation for computer science

170 BINOMIAL COEFFICIENTS
that m = 0; hence the identity to be proved is
& (L)(nik) = (r:s)~ integer n. (5.27)
Let T and s be nonnegative integers; the general case then follows by the
polynomial argument. On the right side, (‘L”) is the number of ways to
choose n people from among r men and s women. On the left, each term Sexist! You menof
the sum is the number of ways to choose k of the men and n - k of the Coned men first.
women. Summing over all k. counts each possibility exactly once.
Much more often than n.ot we use these identities left to right, since that’s
the direction of simplification. But every once in a while it pays to go the
other direction, temporarily making an expression more complicated. When
this works, we’ve usually created a double sum for which we can interchange
the order of summation and then simplify.
Before moving on let’s look at proofs for two more of the identities in
Table 169. It’s easy to prove (5.23); all we need to do is replace the first
binomial coefficient by (,-k-,), then Vandermonde’s (5.22) applies.
The next one, (5.24), is a bit more difficult. We can reduce it to Vandermonde’s
convolution by a sequence of transformations, but we can just
as easily prove it by resorting to the old reliable technique of mathematical
induction. Induction is often the first thing to try when nothing else obvious
jumps out at us, and induction on 1 works just fine here.
For the basis 1 = 0, all terms are zero except when k = -m; so both sides
of the equation are (-l)m(s;m). N ow suppose that the identity holds for all
values less than some fixed 1, where 1 > 0. We can use the addition formula
to replace (,\,) by (,,!,yk) i- (,i-,‘_,) ; th e original sum now breaks into two
sums, each of which can be evaluated by the induction hypothesis:
q (A,;) (“‘I”)‘--‘)“+& (m;;‘l) (s;k)(-l)*
And this simplifies to the right-hand side of (5.24), if we apply the addition
formula once again.
Two things about this derivation are worthy of note. First, we see again
the great convenience of summing over all integers k, not just over a certain
range, because there’s no need to fuss over boundary conditions. Second,
the addition formula works nicely with mathematical induction, because it’s
a recurrence for binomial coefficients. A binomial coefficient whose upper
index is 1 is expressed in terms of two whose upper indices are 1 - 1, and
that’s exactly what we need to apply the induction hypothesis.