Suppose that $P$ is the poset consisting of all subsets of $\{1, 2,....,n\}$ ordered by the relation $a \leq b$ if $a_i \leq b_i$ for all $1 \leq i \leq k$ where $a = \{a_1, a_2,....., a_k\}$, $b = \{b_1, ..... , b_l\}$, and $k \leq l$ where elements are ordered such that $a_1 > a_2 ...$ and $b_1 > b_2 .....$. Prove that $P$ is graded with rank ${n+1} \choose 2$.
My efforts:
It is easy to see there is a maximal chain of length ${n+1} \choose 2$ namely $\phi - \{1\}-\{2\}-....-\{n\}-\{n,1\}-\{n,2\}-....-\{n,n - 1\}-\{n,n-1,1\}-\{n,n-1,2\}-....-\{n,n-1,n-2\}-......-\{n,n-1,n-2,...,1\}$
I don't know how to show every other maximal chain will have same length.
Thanks in advance!!