Question

A partial order <= is defined on the set S = {x,a1,a2,...,an,y} as x<=ai for all i and ai<=y for all i, where n>=1. Number of total orders on the set S which contain partial order <= is
a) 1
b) n
c) n+2
d) n!

Answer 1

We fix only minimal and maximal element, so positions of other n element are
arbitrary.
Number of total orders coincide with number of permutations of set
{1,2,...,n}, since every total order can be interpreted as n-tuple of elements
a_i.
Thus as it is well-known there are n! different permutations of set
{1,2,...,n}. So answer is d)