If `A_(i,j)` be the coefficient of `a^i b^j c^(2010-i-j)` in the expansion of `(a+b+c)^2010`, then

Options

A. (a) `A_(i,i)` is defined for `i ge 1010`
B. (b) `A_(i,j)=A_(j,i)`
C. (c) `A_(2i,3i)` is defined for `i ge 405`
D. (d) `A_(0,1)=2000`

Explanation

To solve the problem, we need to find the coefficient \( A_{i,j} \) of \( a^i b^j c^{2010-i-j} \) in the expansion of \( (a+b+c)^{2010} \).

### Step-by-Step Solution:

1. **Understanding the Expansion**:
The expansion of \( (a+b+c)^{2010} \) can be expressed using the multinomial theorem. The general term in the expansion is given by:
\[ \frac{2010!}{i! j! k!} a^i b^j c^k \]
where \( k = 2010 - i - j \).

2. **Identifying the Coefficient**:
The coefficient \( A_{i,j} \) corresponds to the term where \( a \) is raised to the power \( i \), \( b \) to the power \( j \), and \( c \) to the power \( 2010 - i - j \). Thus, we can write:
\[ A_{i,j} = \frac{2010!}{i! j! (2010 - i - j)!} \]

3. **Conditions for Validity**:
For \( A_{i,j} \) to be defined, the following conditions must be satisfied:
- \( i + j + k = 2010 \) (which is satisfied by the definition of \( k \))
- \( i, j, k \geq 0 \) (which implies \( i \leq 2010 \), \( j \leq 2010 \), and \( 2010 - i - j \geq 0 \) or \( i + j \leq 2010 \))

4. **Analyzing the Options**:
- **Option A**: \( A_{i,i} \) is defined for \( i \geq 1010 \).
- Here, \( A_{i,i} \) would require \( 2i \leq 2010 \), which implies \( i \leq 1005 \). Thus, this option is incorrect.

- **Option B**: \( A_{i,j} = A_{j,i} \).
- From the formula, we see that \( A_{i,j} = \frac{2010!}{i! j! (2010 - i - j)!} \) and \( A_{j,i} = \frac{2010!}{j! i! (2010 - j - i)!} \). Since both expressions are equal, this option is correct.

- **Option C**: \( A_{2i,3i} \) is defined for \( i \geq 405 \).
- For \( A_{2i,3i} \), we need \( 5i \leq 2010 \), which implies \( i \leq 402 \). Thus, this option is incorrect.

- **Option D**: \( A_{0,1} = 2000 \).
- Here, \( A_{0,1} = \frac{2010!}{0! 1! 2009!} = \frac{2010}{1} = 2010 \). Thus, this option is incorrect.

### Conclusion:
The only correct option is **Option B**: \( A_{i,j} = A_{j,i} \).