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.
I n
Binomial coefficients were well known in Asia, many centuries before Pascal was born 1741, but he bad no way to know that.
0
1
1 2 3 4 5 6 7 8 9 10
1
1 3 6 5 6 7 8 9 10
1 4 10 15 21 28 36 45
10 20 35 56 84 120
1 5 15 35 70 126 210
1 6 21 56 126 252
1 7 28 84 210
1 8 36 120
1 9 45
1 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 0
In Italy it’s called Tartaglia’s triangle.
1
12 13 14 1 1 1 1 1 1
=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.
