`""^13C_9-""^12 C_8` is equal to

Options

A. `"^12 C_9`
B. `"^12 C_10`
C. `"^12 C_7`
D. `"^13 C_7`

Answer

To solve the expression \( ^{13}C_9 - ^{12}C_8 \), we can use the combinatorial identity that relates combinations. The identity states that:

\[ ^{n}C_{r} = ^{n-1}C_{r} + ^{n-1}C_{r-1} \]

This means that the number of ways to choose \( r \) items from \( n \) items can be expressed as the sum of the number of ways to choose \( r \) items from \( n-1 \) items and the number of ways to choose \( r-1 \) items from \( n-1 \) items.

### Step-by-step Solution:

1. **Identify the values**: We have \( n = 13 \) and \( r = 9 \) for \( ^{13}C_9 \), and \( n = 12 \) and \( r = 8 \) for \( ^{12}C_8 \).

2. **Apply the identity**: We can rewrite \( ^{13}C_9 \) using the identity:
\[ ^{13}C_9 = ^{12}C_9 + ^{12}C_8 \]

3. **Substitute into the expression**: Now, substituting this back into our expression:
\[ ^{13}C_9 - ^{12}C_8 = (^{12}C_9 + ^{12}C_8) - ^{12}C_8 \]

4. **Simplify the expression**: The \( ^{12}C_8 \) terms cancel out:
\[ ^{13}C_9 - ^{12}C_8 = ^{12}C_9 \]

5. **Final Result**: Thus, we find that:
\[ ^{13}C_9 - ^{12}C_8 = ^{12}C_9 \]

### Conclusion:
The final answer is \( ^{12}C_9 \).