What is the sum of all numbers that occur an odd number of times in rows 0 through 11 of Pascal's triangle?

Details and assumptions
The 0th row of the Pascal triangle is the vertex, which is 1.

You only sum the number once, regardless of the number of times it appears in the triangle.

To find the sum of all numbers that occur an odd number of times in rows 0 through 11 of Pascal's triangle, you can follow these steps:

Step 1: Generate Pascal's triangle up to the 11th row.
Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. Start by generating Pascal's triangle up to the 11th row. The 0th row is 1, so you can write it as:

1
1 1
1 2 1
1 3 3 1
...

Keep generating the subsequent rows until you reach the 11th row.

Step 2: Count the occurrences of each number.
Once you have generated the entire Pascal's triangle up to the 11th row, count the number of occurrences of each value in the triangle. You can use a hash map or dictionary to keep track of the counts. For example, if the number 2 occurs 3 times, you would have `{2: 3}` in your hash map.

Step 3: Sum the numbers that occur an odd number of times.
Finally, iterate through the hash map or dictionary and calculate the sum of all the numbers that occur an odd number of times. Since you only want to include each number once, regardless of the number of occurrences, you can directly sum the keys of the hash map or dictionary where the count is odd.

For example, if your hash map/dictionary looks like `{1: 2, 2: 3, 3: 4}`, you would sum the keys 1 and 3 since they have an odd count. In this case, the sum would be 1 + 3 = 4.

By following these steps, you can find the sum of all numbers that occur an odd number of times in rows 0 through 11 of Pascal's triangle.