Given:
The sum of the binomial coefficients from k = 10 to k = 2006 is given by:
∑(k=10)^(2006)kC10

To find:
The value of (n p) where p is prime.

Solution:

Step 1: Simplifying the given expression

We know that the binomial coefficient nCr can be expressed as nC(r-1) * (n-r+1) / r.

Using this property, we can simplify the given sum as follows:

∑(k=10)^(2006)kC10 = ∑(k=10)^(2006)(kC9 * (k+1) / 10)

Now, let's split the sum into two parts:

Part 1: ∑(k=10)^(2006)kC9
Part 2: ∑(k=10)^(2006)(k+1) / 10

Step 2: Evaluating Part 1

Using the same property as above, we can further simplify Part 1:

∑(k=10)^(2006)kC9 = ∑(k=10)^(2006)(kC8 * (k+1) / 9)

Continuing this process, we can simplify Part 1 as:

∑(k=10)^(2006)kC9 = ∑(k=10)^(2006)(kC0 * (k+1) / 9)

Since kC0 = 1 for all values of k, we can rewrite Part 1 as:

Part 1 = ∑(k=10)^(2006)(k+1) / 9

Step 3: Evaluating Part 2

Using the property of arithmetic series, we can evaluate Part 2 as follows:

Part 2 = ((2006+1) + 10) * ((2006-10+1) / 2) / 10
= (2007 + 10) * (1997 / 2) / 10
= 10017

Step 4: Evaluating the final expression

Now, substituting the values of Part 1 and Part 2 into the original expression:

∑(k=10)^(2006)kC10 = Part 1 * Part 2
= (∑(k=10)^(2006)(k+1) / 9) * 10017

Step 5: Determining the value of (n p)

From the simplified expression above, we can see that the value of (n p) will be equal to the coefficient of n in the final expression. In this case, the coefficient of n is 1.

Therefore, the value of (n p) is 1.

Final Answer: The value of (n p) is 1. Therefore, the correct option is (1) 2017.