What is C( 47,4) `+ ` C ( 51,3) `+` C ( 50,3) `+` C ( 49,3) `+` C ( 48,3) `+` C ( 47,3) equal to ?

Options

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

Answer

To solve the expression \( C(47, 4) + C(51, 3) + C(50, 3) + C(49, 3) + C(48, 3) + C(47, 3) \), we will use the properties of combinations.

### Step-by-Step Solution:

1. **Identify the Combinations**:
We have the following combinations:
- \( C(47, 4) \)
- \( C(51, 3) \)
- \( C(50, 3) \)
- \( C(49, 3) \)
- \( C(48, 3) \)
- \( C(47, 3) \)

2. **Use the Combination Identity**:
Recall the identity:
\[ C(n, r) + C(n, r+1) = C(n+1, r+1) \]
We will apply this identity to combine the terms.

3. **Combine \( C(47, 4) \) and \( C(47, 3) \)**:
Using the identity:
\[ C(47, 4) + C(47, 3) = C(48, 4) \]
Now, our expression becomes:
\[ C(48, 4) + C(51, 3) + C(50, 3) + C(49, 3) + C(48, 3) \]

4. **Combine \( C(48, 3) \) with \( C(49, 3) \)**:
Again using the identity:
\[ C(48, 3) + C(49, 3) = C(49, 4) \]
Now, our expression is:
\[ C(48, 4) + C(51, 3) + C(50, 3) + C(49, 4) \]

5. **Combine \( C(49, 4) \) with \( C(50, 3) \)**:
Using the identity:
\[ C(49, 4) + C(50, 3) = C(50, 4) \]
Now, our expression simplifies to:
\[ C(48, 4) + C(51, 3) + C(50, 4) \]

6. **Combine \( C(50, 4) \) with \( C(51, 3) \)**:
Finally, applying the identity:
\[ C(50, 4) + C(51, 3) = C(51, 4) \]
Thus, we have:
\[ C(52, 4) \]

### Final Result:
The final result of the expression \( C(47, 4) + C(51, 3) + C(50, 3) + C(49, 3) + C(48, 3) + C(47, 3) \) is:
\[ C(52, 4) \]