Introduction:
In this question, we are given a sequence of numbers that are inserted into an empty binary search tree. We need to determine the height of the binary search tree after all the numbers have been inserted.

Binary Search Tree:
A binary search tree is a type of binary tree where the values of all the nodes in the left subtree are less than the value of the root node, and the values of all the nodes in the right subtree are greater than the value of the root node. This property holds true for all the nodes in the tree.

Inserting the Numbers:
Let's go through the given sequence of numbers and insert them into the binary search tree:
1. Insert 10 as the root of the tree.
2. Insert 1 as the left child of 10.
3. Insert 3 as the right child of 1.
4. Insert 5 as the right child of 3.
5. Insert 11 as the right child of 10.
6. Insert 12 as the right child of 11.
7. Insert 6 as the right child of 5.

Understanding the Height:
The height of a binary search tree is defined as the maximum number of edges in any path from the root to a leaf node. In other words, it represents the longest path from the root to a leaf node.

Calculating the Height:
To calculate the height of the binary search tree, we need to find the longest path from the root to a leaf node. Let's analyze the tree:

- The root node is at level 0.
- The left child of the root (1) is at level 1.
- The right child of 1 (3) is at level 2.
- The right child of 3 (5) is at level 3.
- The right child of 5 (6) is at level 4.
- The right child of the root (11) is at level 1.
- The right child of 11 (12) is at level 2.

The longest path from the root to a leaf node is 4, which corresponds to the path: 10 -> 1 -> 3 -> 5 -> 6. Therefore, the height of the binary search tree is 4.

Conclusion:
The height of the binary search tree after inserting the given numbers is 4. The height represents the longest path from the root to a leaf node, which is 4 in this case.