For `r = 0, 1,"…..",10`, let `A_(r),B_(r)`, and `C_(r)` denote, respectively, the coefficient of `x^(r )` in the expansion of `(1+x)^(10), (1+x)^(20)` and `(1+x)^(30)`. Then `sum_(r=1)^(10) A_(r)(B_(10)B_(r ) - C_(10)A_(r ))` is equal to

Options

A. `B_(10) - C_(10)`
B. `A_(1)(B_(10)^(2) - C_(10)A_(10))`
C. 0
D. `C_(10) - B_(10)`

Explanation

To solve the problem, we need to evaluate the expression

\[ \sum_{r=1}^{10} A_r (B_{10} B_r - C_{10} A_r) \]

where \( A_r \), \( B_r \), and \( C_r \) are the coefficients of \( x^r \) in the expansions of \( (1+x)^{10} \), \( (1+x)^{20} \), and \( (1+x)^{30} \) respectively.

### Step-by-step Solution:

1. **Identify the Coefficients**:
- The coefficient \( A_r \) in \( (1+x)^{10} \) is given by \( \binom{10}{r} \).
- The coefficient \( B_r \) in \( (1+x)^{20} \) is given by \( \binom{20}{r} \).
- The coefficient \( C_r \) in \( (1+x)^{30} \) is given by \( \binom{30}{r} \).

2. **Substituting the Coefficients**:
- Substitute \( A_r \), \( B_r \), and \( C_r \) into the sum:
\[ \sum_{r=1}^{10} \binom{10}{r} \left( \binom{20}{10} \binom{20}{r} - \binom{30}{10} \binom{10}{r} \right) \]

3. **Distributing the Sum**:
- Distributing the sum gives:
\[ \sum_{r=1}^{10} \binom{10}{r} \binom{20}{10} \binom{20}{r} - \sum_{r=1}^{10} \binom{10}{r} \binom{30}{10} \binom{10}{r} \]

4. **Evaluating the First Sum**:
- The first sum can be simplified using the binomial theorem:
\[ \binom{20}{10} \sum_{r=1}^{10} \binom{10}{r} \binom{20}{r} = \binom{20}{10} \cdot 2^{10} \]
(since \( \sum_{r=0}^{n} \binom{n}{r} \binom{m}{r} = \binom{n+m}{n} \))

5. **Evaluating the Second Sum**:
- The second sum can also be simplified:
\[ \sum_{r=1}^{10} \binom{10}{r}^2 = \binom{20}{10} \]
(using the identity \( \sum_{r=0}^{n} \binom{n}{r}^2 = \binom{2n}{n} \))

6. **Putting It All Together**:
- Now we have:
\[ \binom{20}{10} \cdot 2^{10} - \binom{30}{10} \cdot \binom{20}{10} \]
- Factor out \( \binom{20}{10} \):
\[ \binom{20}{10} (2^{10} - \binom{30}{10}) \]

7. **Final Result**:
- Thus, the final result is:
\[ \sum_{r=1}^{10} A_r (B_{10} B_r - C_{10} A_r) = \binom{20}{10} (2^{10} - \binom{30}{10}) \]