Formulas for different permutation/combination scenarios I was trying to develop formulas for different permutation/combination scenarios, but I could not sort out last three cases. Please check these following cases -

Unique items


*

*Repetition: no 


*

*Permutation: $_nP_r$

*Combination: $_nC_r$


*Repetition: yes


*

*Permutation: $n^r$

*Combination: $_{n+r-1}C_r$ or $_{n+r-1}C_{n-1}$




Non-unique items


*

*Repetition: no


*

*Permutation: $\displaystyle \frac{n!}{k_1!k_2!\cdots k_n!}$, where $k_1,k_2,\dots k_n$ are numbers of non-unique items

*Combination: ??


*Repetition: yes


*

*Permutation: ??

*Combination: ??


 A: Closely related with your question is a somewhat more general consideration of fundamental counting techniques called 

The twelvefold way 
R.P. Stanley presents in his classic Enumerative combinatorics vol. 1 in section 1.9 the so-called    twelvefold way.  He considers finite  sets   $N$  and $X$, with $|N|=n,  |X|=x$ and counts the  number of different  functions      $f:N\rightarrow  X$   under different situations.



*

*Functions: $f$ may be arbitrary, injective or surjective giving three different possibilities.

*Sets: Elements of $N,X$ may be either distinguishable or indistinguishable resulting in four different possibilities.

Altogether we can consider $3\cdot 4=12$ different situations:
\begin{array}{ll|ccc}
\text{elements }N&\text{elements }X&\quad\text{any }f\quad&\quad\text{injective }f\quad&\quad\text{surjective } f\quad\\
\hline
\text{dist.}&\text{dist.}&x^n\quad&\quad x^{\underline{n}}\quad&x!{n\brace x}\\
\text{indist.}&\text{dist.}&\left(\!\!{x\choose n}\!\!\right)\quad&\quad\binom{x}{n}\quad&\left(\!\!{x\choose n-x}\!\!\right)\quad\\
\text{dist.}&\text{indist.}&\sum_{j=0}^x{n\brace j}\quad&\quad\begin{matrix}1&\text{if }n\leq x\\0&\text{if }n>x\end{matrix}\quad&{n\brace x}\quad\\
\text{indist.}&\text{indist.}&\sum_{j=0}^xp_j(n)\quad&\quad\begin{matrix}1&\text{if }n\leq x\\0&\text{if }n>x\end{matrix}\quad&p_x(n)\quad\\
\end{array}

with
$\qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1)$ the falling factorial of $x$,
$\qquad x!=x(x-1)\cdots 3\cdot2\cdot1$    the factorial of $x$,
$\qquad \binom{x}{n}=\frac{x!}{n!(x-n)!}$ the binomial coefficient $x$ choose $n$,
$\qquad \left(\!\!{x\choose n}\!\!\right)=\binom{x+n-1}{n}$ the number of multisets $x$ multichoose $n$.
$\qquad {n\brace x}$ the Stirling numbers of second kind and 
$\qquad p_x(n)$ the number of partitions of $n$ into $x$ parts.
A presentation in terms of urns and balls can be found here.

[2016-10-15] Add-on:
Some information regarding properties of functions and sets added due to a comment of OP.
A function $f:N\rightarrow X$ is said to be
  
  
*
  
*arbitrary or non-restrictive if there is no specific restriction given
  
*injective or one-to-one if each element of $X$ is the image of at most one element of $N$
  
*surjective or onto if each element of $X$ is the image of at least one element of $N$

Examples: Let's take a look at some functions with respect to these properties:

\begin{array}{l|ccc}
\text{function}&arbitrary&injective&surjective\\
\hline
\\
f:\{1,2,3\}\rightarrow\{a,b,c,d\}&\mathbb{\color{blue}{\text{yes}}}&-&-\\
f(1)=f(2)=c,f(3)=a\\
\\
g:\{1,2,3\}\rightarrow\{a,b,c,d\}&\mathbb{\color{blue}{\text{yes}}}&\mathbb{\color{blue}{\text{yes}}}&-\\
g(1)=d,g(2)=c,g(3)=a\\
\\
h:\{1,2,3,4\}\rightarrow\{a,b,c\}&\mathbb{\color{blue}{\text{yes}}}&-&\mathbb{\color{blue}{\text{yes}}}\\
h(1)=h(4)=c,h(2)=a,h(3)=b\\
\\
i:\{1,2,3,4\}\rightarrow\{a,b,c,d\}&\mathbb{\color{blue}{\text{yes}}}&\mathbb{\color{blue}{\text{yes}}}&\mathbb{\color{blue}{\text{yes}}}\\
i(1)=d,i(2)=c,i(3)=a,i(4)=b\\
\end{array}

Balls and boxes
We think of $N=\{1,2,3\}$ as a set of balls and of $X=\{a,b,c,d\}$ as a set of boxes. A function $f:N\rightarrow X$ is considered as  placing each ball into some box.

We consider four functions $j,k,l,m: N\rightarrow X$ by
  \begin{array}{lclcllcl}
j(1)&=&j(2)&=&a,&\qquad j(3)&=&b\\
k(1)&=&k(3)&=&a,&\qquad k(2)&=&b\\
l(1)&=&l(2)&=&b,&\qquad l(3)&=&d\\
m(2)&=&m(3)&=&b,&\qquad m(1)&=&c\\
\end{array}

Four functions with distinguishable balls and boxes:
                                 
with balls indistinguishable:
                                 
with boxes indistinguishable:
                                 
with balls and boxes indistinguishable:
                                 
