One easily sees that

 since

Also,the proof of which is left to the reader. However,

as can easily be shown by counterexample. (See Problem 19 of this chapter.)

A corresponding notation for products uses the Greek letter pi:

In this notation, n! for can be expressed as
In solving problems stated in terms of the sigma or pi notation, it is sometimes helpful to rewrite the expression in the original notation.
 

Problems for Chapter 6
 
1. Find each of the following:
 
(a) The coefficient of x⁴y¹⁶ in (x + y)²⁰.
 
(b) The coefficient of x⁵ in (1 + x)¹⁵.
 
(c) The coefficient of x³y¹¹ in (2x - y)¹⁴.

Monday, June 8, 1998