The question states that the sum of two numbers is 24, and we need to determine the greatest possible product of these two numbers. Let's break down the problem step by step to find the correct answer.

Step 1: Express the two numbers algebraically
Let's assume the two numbers are x and y. According to the given information, their sum is 24. So, we can write this as an equation:
x + y = 24

Step 2: Express one variable in terms of the other
To find the greatest product, we need to express one variable in terms of the other so that we can maximize the product. Let's solve the equation for y:
y = 24 - x

Step 3: Express the product of the two numbers
The product of the two numbers, p, can be expressed as:
p = x * y

Step 4: Substitute the expression for y in terms of x into the product equation
Now, substitute the expression for y in terms of x into the product equation:
p = x * (24 - x)

Step 5: Simplify the equation
To find the maximum value of p, we need to simplify the equation and express it in the standard form of a quadratic equation (ax^2 + bx + c = 0):
p = 24x - x^2

Step 6: Find the maximum value of the quadratic equation
To find the maximum value of the quadratic equation, we can use the fact that the maximum or minimum value of a quadratic occurs at the vertex. The x-coordinate of the vertex for a quadratic equation in the form ax^2 + bx + c = 0 is given by x = -b/2a. In our equation, a = -1 and b = 24, so the x-coordinate of the vertex is x = -24/(2*(-1)) = 12.

Step 7: Find the maximum product
To find the maximum product, substitute the value of x into the product equation:
p = 12 * (24 - 12) = 12 * 12 = 144

Therefore, the greatest product of the two numbers is 144, which corresponds to option B.