the value of'"" (40)C_(31)+sum_(r=0)'""^(40+r)C_(10-r) is equal to

Options

A. `""^(51)C_(20)`
B. `2""^(50)C_(20)`
C. `2""^(45)C_(15)`
D. none of these

Explanation

To solve the problem, we need to evaluate the expression:

\[ \binom{40}{31} + \sum_{r=0}^{10} \binom{40+r}{10-r} \]

### Step 1: Simplify the Summation

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