Three circles, each with a radius of 10, are mutually tangent to each other. The area enclosed by the three circles can be written as ab√−cπ, where a, b and c are positive integers, and b is not divisible by a square of a prime. What is the value of a+b+c?

To find the area enclosed by the three circles, we can break down the problem into three parts:

1. The area of each circle: The formula to find the area of a circle is A = π * r^2, where A is the area and r is the radius. Given that each circle has a radius of 10, the area of each circle is: A1 = A2 = A3 = π * 10^2.

2. The area of the equilateral triangle formed by connecting the centers of the circles: Since the three circles are mutually tangent to each other, the centers form an equilateral triangle. The formula to find the area of an equilateral triangle is A = (√3/4) * a^2, where A is the area and a is the side length. We need to find the side length of the equilateral triangle.

Let's draw lines from each center to the tangency point between the circles. This creates three congruent triangles, each with a base of 10 (radius) and a height of 10√3 (altitude of an equilateral triangle). Using the Pythagorean theorem, we can find the side length of the equilateral triangle:
side length = 2 * (10^2 - (10√3/2)^2)^0.5 = 2 * (100 - 75)^0.5 = 2 * 25^0.5 = 10√2.

Now we can calculate the area of the equilateral triangle:
A_triangle = (√3/4) * (10√2)^2 = (√3/4) * 100 * 2 = 50√3.

3. The area enclosed by the three circles is the sum of the areas of the three circles minus the area of the equilateral triangle. Let's calculate it:
A_enclosed = A1 + A2 + A3 - A_triangle = 3π * 10^2 - 50√3.

Thus, the area enclosed by the three circles is 3π * 100 - 50√3. Comparing this to the given form ab√−cπ, we can see that a = 3, b = -50, and c = -3. However, we are asked to find positive integers for a, b, and c.

To convert the negative sign into a positive sign, we can multiply the entire expression by -1:
A_enclosed = -3π * 100 + 50√3.

Now, a = 3, b = 50, and c = 3. The sum of a, b, and c is 3 + 50 + 3 = 56.

Therefore, the value of a + b + c is 56.