The coefficient of x^401 in the expansion of (1 + x + x^2 + .......... + x^9)^-1 can be determined using the concept of generating functions and the Binomial Theorem.

Generating Functions:
Generating functions are a powerful tool in combinatorics that allow us to represent a sequence as a power series. In this case, we are interested in finding the coefficient of x^401 in the expansion of the given generating function.

The Binomial Theorem:
The Binomial Theorem states that for any real number a and any non-negative integer n, the expansion of (1 + x)^n can be expressed as a sum of terms of the form C(n, k) * a^(n-k) * x^k, where C(n, k) represents the binomial coefficient.

Steps to find the coefficient of x^401:
To find the coefficient of x^401 in the expansion of (1 + x + x^2 + .......... + x^9)^-1, we can use the concept of generating functions and the Binomial Theorem.

1. Rewrite the given expression as (1 - (-x))^(-1), where -x is the common ratio of the geometric sequence x, x^2, ..., x^9.

2. By applying the Binomial Theorem to (1 - (-x))^(-1), we can express it as a sum of terms of the form C(-1, k) * (-x)^k.

3. Simplify the expression by using the fact that C(-1, k) = (-1)^k * C(k + 1, k).

4. The coefficient of x^401 will be obtained from the term (-1)^k * C(k + 1, k) * (-x)^k where k satisfies the equation 401 = k + 1.

5. Substitute the value of k = 400 in the expression to find the coefficient of x^401.

6. The resulting coefficient will be (-1)^400 * C(401, 400) * (-1)^400 = (-1)^400 * C(401, 400) = C(401, 400).

Final Answer:
The coefficient of x^401 in the expansion of (1 + x + x^2 + .......... + x^9)^-1 is given by C(401, 400).