Let S={1,2,3,…11} and T1,T2…,TN be distinct subsets of S such that |Ti∩Tj|≤2 for all values i≠j.

What is the maximum possible value of N?

(The empty set is a subset of every set.)

232 is right

Thank you :)

but if u don't mind,can u explain it?

To solve this problem, we need to find the maximum possible value of N, which represents the number of distinct subsets of S such that the intersection of any two subsets has at most 2 elements.

Let's consider the subsets with the maximum number of elements first. In this case, the subsets with the maximum number of elements are the subsets with 11 elements (i.e. subsets that contain all the numbers from S).

Now, if we choose two subsets with 11 elements, their intersection will have 11 elements because they contain all the numbers from S. However, we need the intersection of any two subsets to have at most 2 elements. Therefore, we cannot choose more than one subset with 11 elements.

Next, let's consider the subsets with 10 elements. If two subsets with 10 elements have an intersection of 3 or more elements, we violate the condition. So, we can choose at most one subset with 10 elements.

Similarly, for subsets with 9 elements, their intersection can have at most 4 elements. So, we can choose at most one subset with 9 elements.

Continuing this pattern, we can see that for subsets with 8 elements, the intersection can have at most 5 elements. So, we can choose at most one subset with 8 elements.

Finally, for subsets with 7 or fewer elements, the intersections can have at most 6 or more elements, which do not satisfy the condition. Therefore, we cannot choose any subsets with 7 or fewer elements.

Based on these observations, the maximum possible value of N is the sum of the maximum number of possible subsets for each size:

1 subset with 11 elements +
1 subset with 10 elements +
1 subset with 9 elements +
1 subset with 8 elements = 4

Hence, the maximum possible value of N is 4.