If `a+b+c=24`, then the maximum value of abc is :

Options

A. 215
B. 512
C. 125
D. 576

Answer

To find the maximum value of \( abc \) given that \( a + b + c = 24 \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Understanding the AM-GM Inequality**:
The AM-GM inequality states that for any non-negative numbers \( a, b, c \):
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
This means that the arithmetic mean of the numbers is greater than or equal to the geometric mean.

2. **Applying the Given Condition**:
Since we know that \( a + b + c = 24 \), we can substitute this into the inequality:
\[ \frac{24}{3} \geq \sqrt[3]{abc} \]

3. **Calculating the Arithmetic Mean**:
Simplifying the left side:
\[ 8 \geq \sqrt[3]{abc} \]

4. **Cubing Both Sides**:
To eliminate the cube root, we cube both sides of the inequality:
\[ 8^3 \geq abc \]
Calculating \( 8^3 \):
\[ 8^3 = 512 \]
Thus, we have:
\[ abc \leq 512 \]

5. **Finding When Equality Holds**:
The equality in the AM-GM inequality holds when all the numbers are equal. Therefore, we set:
\[ a = b = c \]
Since \( a + b + c = 24 \), we have:
\[ 3a = 24 \implies a = 8 \]
Thus, \( a = b = c = 8 \).

6. **Calculating the Maximum Value of \( abc \)**:
Now substituting back to find \( abc \):
\[ abc = 8 \times 8 \times 8 = 512 \]

### Conclusion:
The maximum value of \( abc \) given \( a + b + c = 24 \) is \( \boxed{512} \).