The value of `sum_(r=0)^(40) r""^(40)C_(r)""^(30)C_(r)` is

Options

A. `40.^(69)C_(29)`
B. `40.^(70)C_(30)`
C. `.^(60)C_(29)`
D. `.^(70)C_(30)`

Explanation

To solve the problem, we need to find the value of the summation:

\[ \sum_{r=0}^{40} r \cdot \binom{40}{r} \cdot \binom{30}{r} \]

### Step-by-Step Solution:

1. **Rewrite the term**: We start by rewriting \( r \cdot \binom{40}{r} \):
\[ r \cdot \binom{40}{r} = 40 \cdot \binom{39}{r-1} \]
This is because \( r \cdot \binom{n}{r} = n \cdot \binom{n-1}{r-1} \).

2. **Substituting into the summation**: Now we can substitute this into our summation:
\[ \sum_{r=0}^{40} r \cdot \binom{40}{r} \cdot \binom{30}{r} = \sum_{r=0}^{40} 40 \cdot \binom{39}{r-1} \cdot \binom{30}{r} \]

3. **Changing the index of summation**: To simplify the summation, we can change the index of summation by letting \( k = r - 1 \). Thus, when \( r = 0 \), \( k = -1 \) (which we can ignore), and when \( r = 40 \), \( k = 39 \):
\[ = 40 \cdot \sum_{k=0}^{39} \binom{39}{k} \cdot \binom{30}{k+1} \]

4. **Using the Hockey Stick Identity**: The summation can be simplified using the Hockey Stick Identity:
\[ \sum_{k=0}^{n} \binom{r}{k} \cdot \binom{s}{n-k} = \binom{r+s}{n} \]
In our case, we can apply this identity:
\[ \sum_{k=0}^{39} \binom{39}{k} \cdot \binom{30}{k+1} = \binom{39 + 30}{40} = \binom{69}{40} \]

5. **Final Calculation**: Now substituting back:
\[ = 40 \cdot \binom{69}{40} \]

### Final Answer:
Thus, the value of the summation is:
\[ 40 \cdot \binom{69}{40} \]