I'm having some trouble in my descrete mathematics class, please help me, thanks?

1. Let A be a set containing 27 elements and B a set containing 11 elements.


(a) How many functions are there from A to B?


(b) How many injective functions from A to B?


(c) How many surjective functions from A to B?





2. How many positive integers less than 3000 are divisible by 11?





3. How many solutions are there to the inequality:


z1 + z2 + z3 is less than or equal to 23


where z1, z2, and z3 are non-negative integers?


(Hint: Introduce a 4th variable z4 and take a look at the


equation z1 + z2 + z3 + z4 = 23)





4. How many permutations of the letters A,B,C,D,E,F,G,H contain


-the string DEG and HAD?

I'm having some trouble in my descrete mathematics class, please help me, thanks?
Here are the first two. I hope this helps some!





1. (a) If x denotes any of the 27 elements in set A, then f(x) can be any of the 11 elements of set B. This means there are 11^27 different ways to construct a function from A to B.


(b) There are no injective functions from A to B. Since B has fewer elements than A, there's no way to map all the elements of A to distinct elements of B.


(c) Constructing a surjective function from A onto B amounts to choosing which 11 elements of A will be mapped onto B, and choosing which one maps to each element of B. The number of different ways this can be done is just the number of permutations of 27 objects taken 11 at a time, commonly written as P(27, 11). This is 27!/(27 - 11)! or 27!/16!.





2. Every 11th positive integer, starting with 11, is divisible by 11. The number of positive integers that are less than or equal to any positive number n is just the greatest integer in n/11. (That is, you divide n by 11, and discard any remainder so that what's left is an integer.)





Saying a positive integer is less than 3000 is the same as saying it's less than or equal to 2999, so the answer is just the greatest integer in 2999/11, namely 272.