may be constructed in three dimensions, corresponding to the volume (x + y)3 of a cube of side x + y; it may indeed be decomposed as
(2.239)
i.e. the sum of volume x2 y of each of three parallelipipeds of sides x, x, and y, to the volume xy2 of each of three parallelipipeds of sides x, y, and y, and finally to the volume y3 of a cube of side y – besides being directly obtainable from Eq. (2.236) after setting n = 3.
The binomial coefficients in Eq. (2.236), of the form
and supports the entries of Pascal’s triangle – denoted as Table 2.1. Careful inspection of this table indicates that the outermost values are always unity, whereas every two consecutive numbers in a given row add up to the value placed in between at the next row. In fact, Eq. (2.240) allows one to write
(2.241)
where factoring n! coupled with elimination of inner parenthesis give rise to
(2.242)
the factorials in denominator may, in turn, be rewritten as
(2.243)
based on their definition, thus allowing further factoring out of (k − 1)! and (n − k)! as
Upon lumping the two factors still in parenthesis, Eq. (2.244) becomes
(2.245)
that degenerates to
(2.246)
the outstanding factors may, in turn, be lumped with the existing factorials to yield
(2.247)
– equivalent, in view of Eq. (2.240), to
thus confirming the initial suggestion and being frequently known as Pascal’s rule.
Table 2.1 Pascal’s triangle encompassing coefficients of power of binomial,
| n | |||||||||||||||||||||
| 0 | 1 | ||||||||||||||||||||
| 1 | 1 | 1 | |||||||||||||||||||
| 2 |
|
