F is a set of all subsets of set {1,2,3,..100} with atleast 50 numbers without disturbing the order.
Now the, smallest such possible subsets will be of length 50 {1,2,..50}, {1,3,4,5,..51} ...till {51,52,..100}.
Set F will contain such subsets of length 50 to 100.
Now, with respect to the $\subseteq$, {1,2,..50} will be subset of {1,2,..51}.
{1,2,..51} will be subset of {1,2,..52}.
Like this we can go till {1,2,..99} will be a subset of {1,2,..100}. The chain from {1,2,..50} to {1,2,..100} will be of size 51. Hence, 51.