Let $ \{a_k\}$ be a sequence of integers such that $ a_1 \equal{} 1$ and $ a_{m \plus{} n} \equal{} a_m \plus{} a_n \plus{} mn$, for all positive integers $ m$ and $ n$. Then $ a_{12}$ is

$ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 67 \qquad \textbf{(D)}\ 78 \qquad \textbf{(E)}\ 89$
1. We start with the given sequence definition and initial condition:
   \[
   a_1 = 1
   \]
   \[
   a_{m+n} = a_m + a_n + mn \quad \text{for all positive integers } m \text{ and } n
   \]

2. We need to find \(a_{12}\). To do this, we will first compute the initial terms of the sequence to identify a pattern.

3. Compute \(a_2\):
   \[
   a_2 = a_1 + a_1 + 1 \cdot 1 = 1 + 1 + 1 = 3
   \]

4. Compute \(a_3\):
   \[
   a_3 = a_2 + a_1 + 2 \cdot 1 = 3 + 1 + 2 = 6
   \]

5. Compute \(a_4\):
   \[
   a_4 = a_3 + a_1 + 3 \cdot 1 = 6 + 1 + 3 = 10
   \]

6. Compute \(a_5\):
   \[
   a_5 = a_4 + a_1 + 4 \cdot 1 = 10 + 1 + 4 = 15
   \]

7. Compute \(a_6\):
   \[
   a_6 = a_5 + a_1 + 5 \cdot 1 = 15 + 1 + 5 = 21
   \]

8. Compute \(a_7\):
   \[
   a_7 = a_6 + a_1 + 6 \cdot 1 = 21 + 1 + 6 = 28
   \]

9. Compute \(a_8\):
   \[
   a_8 = a_7 + a_1 + 7 \cdot 1 = 28 + 1 + 7 = 36
   \]

10. Compute \(a_9\):
    \[
    a_9 = a_8 + a_1 + 8 \cdot 1 = 36 + 1 + 8 = 45
    \]

11. Compute \(a_{10}\):
    \[
    a_{10} = a_9 + a_1 + 9 \cdot 1 = 45 + 1 + 9 = 55
    \]

12. Compute \(a_{11}\):
    \[
    a_{11} = a_{10} + a_1 + 10 \cdot 1 = 55 + 1 + 10 = 66
    \]

13. Compute \(a_{12}\):
    \[
    a_{12} = a_{11} + a_1 + 11 \cdot 1 = 66 + 1 + 11 = 78
    \]

14. We observe that the sequence \(a_k\) follows the pattern of triangular numbers, which can be expressed as:
    \[
    a_k = \frac{k(k+1)}{2}
    \]

15. Verify the formula for \(k = 12\):
    \[
    a_{12} = \frac{12 \cdot 13}{2} = 78
    \]

The final answer is \(\boxed{78}\).