What is `.^(47)C_(4)+.^(51)C_(3)+ underset(j=2)overset(5)sum.^(52-j)C_(3)` equal to ?

Options

A. `""^(52)C_(4)`
B. `""^(52)C_(3)`
C. `""^(51)C_(4)`
D. `""^(53)C_(4)`

Explanation

To solve the expression \( \binom{47}{4} + \binom{51}{3} + \sum_{j=2}^{5} \binom{52-j}{3} \), we will break it down step by step.

### Step 1: Rewrite the summation
The summation can be expanded as follows:
\[ \sum_{j=2}^{5} \binom{52-j}{3} = \binom{50}{3} + \binom{49}{3} + \binom{48}{3} + \binom{47}{3} \]

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