The number of $$4$$ digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set $${1, 2, 3}$$ is____________.

The recurrence relation capturing the optional execution time of the Towers of Hanoi problem with $$n$$ discs is

In a class of 200 students, 125 students have taken Programming Language course, 85 students have taken Data Structures course, 65 student have taken Computer Organization coures; 50 students have taken both Programming Language and Data Structures, 35 students have taken both Programming Languages and Computer Organization; 30 students have taken both Data Structures and Computer Organization; 15 students have taken all the three course.

How many students have not taken any of the three courses?

Let $$A$$ be a sequence of $$8$$ distinct integers sorted in ascending order. How many distinct pairs of sequence, $$B$$ and $$C$$ are there such that
i) Each is sorted in ascending order.
ii) $$B$$ has $$5$$ and $$C$$ has $$3$$ elements, and
iii) The result of merging $$B$$ $$C$$ gives $$A$$?