Number of possible way to select subset WITHOUT order and with/without replacement for repeated entries What is the number of ways that I can select subset (k) from larger set (n) where the order is NOT important and without replacement (I can't use the element more than once).
For example I have the set A={1,2,2} and I need to select only two number from the set. How many possibility are there?
will, manually its {(1,2),(2,2)} so the answer is 2. How to get three directly without listing the possibilities???
Second thing, what if I have the same example but with replacement (I can use the element more than once)>>>> the answer is {(1,1),(1,2),(2,2)}, so 3 possibilities, How to get 3 directly ??
 A: First of all, I'd like to make some important distinctions:

A set is a collection of non-repeated elements without order
A multiset is a collection of elements without order
A tuple is a finite ordered list of elements

It's important to bear in mind the distinction between these and to know which one you need.
For example, in your question, it's clear you're referring to multisets rather than sets, but you aren't referring to tuples for the subsets you're taking as you want the order not to be important.

Partial answer for the last question:
If you allow replacement, then you may as well consider sets rather than multisets as $\{A,B,B\}$ will give the same result as $\{A,B\}$. All you're interested in is the number of distinct elements in this case.
Therefore the question can be rephrased as: How many multisets are there of $k$ elements such that all elements belong a fixed set $S$ of size $n$?
The solution to this is $n+k-1\choose n$

Explanation:
This uses the solution to the Stars and Bars problem. There are $k$ stars and $n-1$ bars. To see how, let us consider the number of times each element in $S$ appears. Each element can appear between $0$ and $k$ times, and the sum of these appearances for each element must be $k$.
In other words, we're interested in how many ways $k$ can be split up between 'jars' which represent each element of $S$.
Example:
Let $S=\{A,B,C,D,E\}$ and $k=5$. Some possible multisets may be $\{A,A,A,C,D\}$ and $\{B,B,D,D,D\}$. This can be represented with ★★★||★|★| and |★★||★★★| respectively.
The stars represent the number of appearances of a certain element, and the bars represent the dividers between each letter. For the first example:
$$\underbrace{★★★}_{\text{Appearances of: }A}|\underbrace{}_B|\underbrace{★}_C|\underbrace{★}_D|\underbrace{}_E$$

