Mathematicians I NEED you to carefully explain these problems?

Question

1) How many di¤erent four-digit numbers are there if
(a) repetition of digits is allowed.
(b) repetition of digits is not allowed.

2). Find n(S) if we know that n(T) = 8; n(T U S) = 17; and n(T n S) = 5.

Answer 1

#1a: The total number of four digit numbers there are is the number of distinct permutations of four of the digits 0-9, subject to the restriction that the first digit is nonzero (we do want a four-digit number after all). Therefore, there are 9 possibilities for the first digit. If repetition is allowed, we have 10 possibilities for each of the digits 2-4, for a total of 9*10*10*10 or 9000 distinct four-digit numbers.

#1b: Since repetition is not allowed, whatever digit was used in the first position will no longer be available for the others, so there are only 9 possibilities for the second digit. And since you can't reuse any of the first two possibilities, there are only 8 possibilities for the thrid digit, and likewise 7 for the fourth. So here are 9*9*8*7 or 4536 distinct four-digit numbers where no two digits repeat.

#2: You are asked to find the number of things in S, given the number of things in T, T ∪ S, and T ∩ S. Now, if T and S were disjoint sets, n(T ∪ S) would be equal to the sum of n(T) and n(S). But since they are not, we must write this sum in terms of sets that are disjoint. We note that T\S, S\T, and T ∩ S are all disjoint sets. Further, (T\S) ∪ (T ∩ S) is precisely T, and these are disjoint sets, so n(T\S) + n(T ∩ S) = n(T), or n(T\S) = n(T) - n(T ∩ S). By the same logic, n(S\T) = n(S) - n(T ∩ S). So since T\S, S\T, and T ∩ S are disjiont sets whose union is T ∪ S, we have:

n(T ∪ S) = n(T\S) + n(S\T) + n(T ∩ S)
Substituting the values for n(T\S) and n(S\T):
n(T ∪ S) = (n(T) - n(T ∩ S))+ (n(S) - n(T ∩ S)) + n(T ∩ S)
Simplifying:
n(T ∪ S) = n(T) + n(S) - n(T ∩ S)
Rearranging to put n(S) on one side:
n(S) = n(T ∪ S) + n(T ∩ S) - n(T)
Now substituting the known values for those three sets:
n(S) = 17 + 5 - 8 = 14

And you are done.

Answer 2

1a) 9*10*10*10=9000
1b) 9*9*8*7=please calculate

2 ) isTnS= Tand S
n(T)+n(S)-n(TandS)=n(TorS)