The value of `""^(50)C_(4)+sum_(r=1)^(6)""^(56-r)C_(3)`, is

Options

A. `""^(56)C_(4)`
B. `""^(56)C_(3)`
C. `""^(55)C_(3)`
D. `""^(55)C_(4)`

Explanation

To solve the problem \( \binom{50}{4} + \sum_{r=1}^{6} \binom{56-r}{3} \), we will break it down into manageable steps.

### Step 1: Understand the Summation
The summation \( \sum_{r=1}^{6} \binom{56-r}{3} \) can be expanded for each value of \( r \) from 1 to 6:
- For \( r = 1 \): \( \binom{55}{3} \)
- For \( r = 2 \): \( \binom{54}{3} \)
- For \( r = 3 \): \( \binom{53}{3} \)
- For \( r = 4 \): \( \binom{52}{3} \)
...