The value of `C(47,4)+C(51,3)+C(50,3)+C(49,3)+C(48,3)+C(47,3)` is equal to

Options

A. C(47,4)
B. C(52,5)
C. C(52,4)
D. C(47,5)

Answer

To solve the problem \( C(47,4) + C(51,3) + C(50,3) + C(49,3) + C(48,3) + C(47,3) \), we will use the property of combinations that states:

\[ C(n, r) + C(n, r + 1) = C(n + 1, r + 1) \]

### Step-by-Step Solution:

1. **Identify the terms**: We have the terms \( C(47, 4) \), \( C(51, 3) \), \( C(50, 3) \), \( C(49, 3) \), \( C(48, 3) \), and \( C(47, 3) \).

2. **Group terms**: We can group \( C(47, 4) \) with \( C(47, 3) \) and the rest of the terms together:
\[ C(47, 4) + C(47, 3) + C(51, 3) + C(50, 3) + C(49, 3) + C(48, 3) \]

3. **Apply the combination property**: Start with \( C(47, 4) + C(47, 3) \):
\[ C(47, 4) + C(47, 3) = C(48, 4) \]

4. **Rewrite the expression**: Now we can rewrite the expression as:
\[ C(48, 4) + C(51, 3) + C(50, 3) + C(49, 3) + C(48, 3) \]

5. **Combine \( C(51, 3) + C(50, 3) + C(49, 3) + C(48, 3) \)**: We can combine these using the same property:
\[ C(51, 3) + C(50, 3) + C(49, 3) + C(48, 3) = C(52, 4) \]

6. **Final combination**: Now we have:
\[ C(48, 4) + C(52, 4) \]
We can combine these:
\[ C(48, 4) + C(48, 4) = C(49, 5) \]

7. **Final result**: The final result is:
\[ C(52, 4) \]

### Conclusion:
Thus, the value of \( C(47,4) + C(51,3) + C(50,3) + C(49,3) + C(48,3) + C(47,3) \) is equal to \( C(52, 4) \).