**Main Problem**: For an integer , let $S_{n}$ denote the sum of the products of the integers from 1 to n taken three at a time. Then, what is the value of ?

**First Hint: **Consider the terms in the expansion of .

**Second Hint: **Classify these terms according to the number of times they occur.

**Solution: **There are totally 1000 terms in the expansion of

. Terms of the form with i,j,k all distinct appear 6 times each. We are interested in the sum of these terms each such term considered only once. Then, there are terms of the form with

. Each such product appears three times. Finally, there are terms of the form . Each such term appears once only. Hence,

**equation I**

where and and

.

Using the well-known formulae given below:

**equation II**

**equation III**

A and C come out to be respectively and .

To calculate B, note that if we add to B, products of the form , then we get all the terms of the product

.

The first factor equals using the formula

**equation IV**

Hence, . Putting these values in equation I, we get

.

*Please do send your comments, suggestions, etc. *

More later

Nalin Pithwa

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