**Problem:**

If 1, are the nth roots of unity, then find the value of .

**Solution:**

**Learning to think: **

Compare it with what we know from our higher algebra — suppose we have to multiply out:

. We know it is equal to the following:

If we examine the way in which the partial products are formed, we see that

(1) the term is formed by taking the letter x out of each of the factors.

(2) the terms involving are formed by taking the letter x out of any three factors, in every possible way, and one of the letters a, b, c, d out of the remaining factor

(3) the terms involving are formed by taking the letter x out of any two factors, in every possible way, and two of the letters a, b, c, d out of the remaining factors

(4) the terms involving x are formed by taking the letter x out of any one factor, and three out of the letters a, b, c, d out of the remaining factors.

(5) the term independent of x is the product of all the letters a, b, c, d.

**Further hint: **

relate the above to sum of binomial coefficients.

and, you are almost done.

More later,

Nalin Pithwa