CIIM 2019 Problem 2

Consider the set $$\{0, 1\}^n = \{X = (x_1, x_2,\dots , x_n) : x_i \in \{0, 1\}, 1 \leq i \leq n\}.$$ We say that $X > Y$ if $X \neq Y$ and the following $n$ inequalities are satisfy $$x_1 \geq y_1, x_1 + x_2 \geq y_1 + y_2,\dots , x_1 + x_2 + \cdots + x_n \geq y_1 + y_2 + \cdots + y_n.$$ We define a chain of length $k$ as a subset ${Z_1,\dots , Z_k} \subseteq \{0, 1\}^n$ of distinct elements such that $Z_1 > Z_2 > \cdots > Z_k.$ Determine the lenght of longest chain in $\{0,1\}^n$ .