Let S = \{ 1, 2, 3, \ldots 12\} and T_1, T_2, \ldots T_a be subsets of S such that T_i \not \subset T_j \, \forall i \neq j . What is the maximum possible value of a?

To find the maximum possible value of a, we need to determine how many subsets T_i can be created such that no subset T_i is a subset of another subset T_j.

Let's start by considering the element 1. We have two options:
Option 1: Include 1 in T_i.
Option 2: Exclude 1 from T_i.

If we include 1 in T_i, we cannot include any other elements that are subsets of T_i that include 1, as it violates the condition T_i not being a subset of T_j for i ≠ j. This means we have one subset T_i containing only 1.
If we exclude 1 from T_i, then we have 2^(n-1) subsets that can be created from the remaining elements, where n is the total number of elements. In our case, n is 11 since we excluded 1. So, we have 2^(11-1) = 2^10 = 1024 subsets that can be created without including 1.

Now, let's move on to the remaining elements 2, 3, ..., 12.
For each element, we have two options: include it or exclude it.
For each inclusion, we have eliminated some subsets because they would be a subset of another subset that includes the current element.
For each exclusion, we can still create the same number of subsets as before, as the exclusion of an element doesn't affect the subsets we previously counted.

Considering all the elements from 2 to 12, each with two options, we have a total number of subsets without including any of the elements from 2 to 12 as: 2^11 * 2^10 * 2^9 * ... * 2^1.

Therefore, the maximum possible value of a is the number of subsets created without including any of the elements repeated as: 2^11 * 2^10 * 2^9 * ... * 2^1 = 2^(11 + 10 + 9 + ... + 1).

The sum of the sequence 11 + 10 + 9 + ... + 1 can be calculated using the formula for the sum of an arithmetic series:
Sum = (n/2)(first term + last term)
In our case, n = 11, first term = 1, and last term = 11. Substituting into the formula, we get:
Sum = (11/2)(1 + 11) = 66.

Substituting the sum back into the expression for the maximum possible value of a, we have:
Maximum possible value of a = 2^66.

Therefore, the maximum possible value of a is 2^66.