Hi,

Consider all tuples of 5 (ordered, not repeated) natural numbers, each between 1 and 50.

$S=\{(c_1,c_2,c_3,c_4,c_5) \in N^5 / \forall i \in [1,5], c_i \in [1, 50] \wedge \forall i,j \in [1,5], i < j => c_i < c_j\}$

I'd like to know the cardinality of the subset of all those elements, with a "Hamming" distance equal or less than 3 to any other tuple. That is, I'm looking for the minimum subset ensuring at least 2 numbers of any other tuple are present in an element of said subset.

$S_3 = \{t \in S / \forall s \in S, \exists t / d_h(t,s) \leq 3\}$

where $d_h$ refers to a non-rigorous definition of Hamming distance: differences between tuples, regardless of the position.

So, besides my likely errors in the definitions (I don't really know how to describe $S_3$ properly, apologies), what I'm looking for is $|S_3|$.