Definition: A $p$-chain is a subset of $p$-simplicies in a simplicial complex $K$.
Let $T$ be a tetrahedron - a 3 - simplex. It is trivial to note that the faces of a tetrahedron are 2 - simplex; in this case, the number of 2 - simplex is four.
Indeed, each face of a tetrahedron is a convex hull of some non - empty subset of the $(3+1)$ vertices that defines the 3 - simplex (or tetrahedron) and so each face is a subset of the tetrahedron and so belongs to a simplicial complex.
There are $2^{4}$ distinct 2-chains and $2^{6}$ distinct 1-chains. How do I observe this?
You are counting the number of subsets of the set of all subsets of each fixed size.
This will work for the analogues of the tetrahedron in any dimension: the simplicial complex $K_n$ with $n+1$ vertices every subset of which defines a face. For each $p$ there are $\binom{n+1}{p}$ faces of dimension $p$ and $$ 2^\binom{n+1}{p} $$ $p$-chains.