Explanation:
To find the greatest coefficient in the expansion of (1 + x)^12, we can use the binomial theorem. The binomial theorem states that for any positive integer n, the expansion of (a + b)^n can be written as:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n

where C(n, k) represents the binomial coefficient, also known as "n choose k".

In this case, we have (1 + x)^12, so a = 1 and b = x. Plugging these values into the binomial theorem formula, we get:

(1 + x)^12 = C(12, 0) * 1^12 * x^0 + C(12, 1) * 1^11 * x^1 + C(12, 2) * 1^10 * x^2 + ... + C(12, 11) * 1^1 * x^11 + C(12, 12) * 1^0 * x^12

Simplifying this expression, we have:

(1 + x)^12 = C(12, 0) + C(12, 1) * x + C(12, 2) * x^2 + ... + C(12, 11) * x^11 + C(12, 12) * x^12

To find the greatest coefficient, we need to determine which term has the highest coefficient. The coefficient of each term is given by the binomial coefficient C(12, k), where k represents the power of x.

Comparing Coefficients:
In order to find the greatest coefficient, we need to compare the binomial coefficients for each term. The binomial coefficient C(12, k) can be calculated using the formula:

C(n, k) = n! / (k! * (n - k)!)

where "!" represents the factorial operation.

Let's calculate the binomial coefficients for the given options:

a) C(12, 4) = 12! / (4! * (12 - 4)!) = 12! / (4! * 8!) = 495
b) C(12, 6) = 12! / (6! * (12 - 6)!) = 12! / (6! * 6!) = 924
c) C(12, 5) = 12! / (5! * (12 - 5)!) = 12! / (5! * 7!) = 792

Comparing the coefficients, we can see that option B has the greatest coefficient, which is 924. Therefore, the correct answer is option B) C(12, 6).