please see attached files

Solution
Question 1:
First counting principle applies to independent tasks and choices. It
states the total number of possible ways to do an independent task or
to make choices.
So if the total number of ways to accomplish first part of task is n and
total ways of doing second part is m then the ways of completing the
task is n*m , This is also multiplicative rules.
Onto function from set A to set B
A function f from A to B is called onto if for all b in B there is an a in A
such that f (a) = b. This means that all elements present in the set B are
used.
For example y= x3 is an onto function since the whole y axis is mapped
to some x.
Kleen closure of any language is set containing all possible strings made
from the elements of L. So if A* is kleen closure of A then it contains all
possible strings that can be formed from A.
Hence * is the kleen closure of A
Permutations starting with 3
Fix 3 at starting position then free slots = 8
Hence permutations P1= 8! = 40 320
Permutations ending with 7
Fix 7 at ending position then free slots = 8
Hence permutations P2= 8! = 40 320
Permutations starting with 3 and ending with 7 = 5040
These are present in both P1 and P2
Hence total permutation = P1 + P2 -5040 = 75 600
Select any three alphabets from 26
This can be done in 26C3 =2600
Now there will be a single permutation such that three letters are in
order.
So total number of ways is 2600
For example if three letters chosen are a,z,k
Then there is...