In Chapter 1 we definedas the coefficient of a^n-rb^r in the expansion of (a + b)ⁿ, and tabulated these coefficients in the arrangement of the Pascal Triangle:
 

n	Coefficients of (a + b)n
0								1
1							1		1
2						1		2		1
3					1		3		3		1
4				1		4		6		4		1
5			1		5		10		10		5		1
6		1		6		15		20		15		6		1
...	.		.		.		.		.		.		.		.

We then observed that this array is bordered with 1's; that is,  for n = 0, 1, 2, ... . We also noted that each number inside the border of 1's is the sum of the two closest numbers on the previous line. This property may be expressed in the form

This formula provides an efficient method of generating successive lines of the Pascal Triangle, but the method is not the best one if we want only the value of a single binomial coefficient for a large n, such as  We therefore seek a more direct approach.

It is clear that the binomial coefficients in a diagonal adjacent to a diagonal of 1's are the numbers 1, 2, 3, ... ; that is, Now let us consider the ratios of binomial coefficients to the previous ones on the same row. For n = 4, these ratios are:

(2)             4/1, 6/4 = 3/2, 4/6 = 2/3, 1/4.

(3)        5/1, 10/5 = 2, 10/10 = 1, 5/10 = 1/2, 1/5.

Monday, June 22, 1998