Question

The value of the expression \( {}^{47} C_4 + \sum_{j=1}^{5} {}^{52-j} C_3 \) is:

• A. \( {}^{52} C_3 \)
• B. \( {}^{51} C_4 \)
• C. \( {}^{52} C_4 \)
• D. \( {}^{51} C_3 \)

Hint:

Use the hockey-stick identity \( \sum_{m=k}^{n} {}^m C_k = {}^{n+1} C_{k+1} - {}^k C_{k+1} \) to simplify sums of combinations with consecutive indices.