Concrete mathematics : a foundation for computer science

Binomial coefficients
were well known
in Asia, many centuries
before Pascal
was born 1741, but
he bad no way to
know that.
In Italy it’s called
Tartaglia’s triangle.
5.1 BASIC IDENTITIES 155
Before getting to the identities that we will use to tame binomial coefficients,
let’s take a peek at some small values. The numbers in Table 155 form
the beginning of Pascal’s triangle, named after Blaise Pascal (1623-1662)
Table 155 Pascal’s triangle.
n
0
1
2
3I4
5
6
7
8
9
10
1
1 1
12 1
13 3 1
14 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
because he wrote an influential treatise about them [227]. The empty entries
in this table are actually O’s, because of a zero in the numerator of (5.1); for
example, (l) = ( 1.0)/(2.1) = 0. These entries have been left blank simply to
help emphasize the rest of the table.
It’s worthwhile to memorize formulas for the first three columns,
r
0
=I, (;)=?., (;)2g;
these hold for arbitrary reals. (Recall that (“T’) = in(n + 1) is the formula
we derived for triangular numbers in Chapter 1; triangular numbers are conspicuously
present in the (;) column of Table 155.) It’s also a good idea to
memorize the first five rows or so of Pascal’s triangle, so that when the pattern
1, 4, 6, 4, 1 appears in some problem we will have a clue that binomial
coefficients probably lurk nearby.
The numbers in Pascal’s triangle satisfy, practically speaking, infinitely
many identities, so it’s not too surprising that we can find some surprising
relationships by looking closely. For example, there’s a curious “hexagon
property,” illustrated by the six numbers 56, 28, 36, 120, 210, 126 that surround
84 in the lower right portion of Table 155. Both ways of multiplying
alternate numbers from this hexagon give the same product: 56.36.210 =
28.120.126 = 423360. The same thing holds if we extract such a hexagon
from any other part of Pascal’s triangle.