Finding the number of ordered pairs of integers (Discrete Maths) Let $k$ and $n$ be positive integers such that $k\le n$
(i) How many ordered sequences of integers ($a_{1}$,$a_{2}$,...$a_{k}$) are there such that $a_{1}$,$a_{2}$,...$a_{k}$$\in $(1,2...n)
(ii) How many ordered sequences of integers ($a_{1}$,$a_{2}$,...$a_{k}$) are there such that $a_{1}$,$a_{2}$,...$a_{k}$$\in $(1,2...n) and $a_{1}$,$a_{2}$,...$a_{k}$ are pairwise distinct?
(iii)How many ordered sequences of integers ($a_{1}$,$a_{2}$,...$a_{k}$) are there such that $a_{1}$,$a_{2}$,...$a_{k}$$\in $(1,2...n) and $a_{1}$,$a_{2}$,...$a_{k}$ contain only one ordered pair?
(iv))How many ordered sequences of integers ($a_{1}$,$a_{2}$,...$a_{k}$) are there such that $a_{1}$,$a_{2}$,...$a_{k}$$\in $(1,2...n) and $a_{1}\lt a_{2}...\lt a_{k} $ ?
(v))How many ordered sequences of integers ($a_{1}$,$a_{2}$,...$a_{k}$) are there such that $a_{1}$,$a_{2}$,...$a_{k}$$\in $(1,2...n) and $a_{1}\le a_{2}...\le a_{k} $ ?
What i tried
(ii) pairwise distinct means that the integers can be group onto a pair with each pair being different from one another, thus we are group indistinguishable integers into distinguishable pair. Thus there are $k^{n}$ ways.
(iii) Ordered pair have got something to do with permutations so i think it is $k$ permutate $n$
(iv) While this part means that the integers must be arranged in order form the smallest to the biggest
Im unsure of how to do these questions. Could anyone explain. Thanks
 A: (i) There are $n$ choices for the first element, and for every such choice there are $n$ choices for the second element, and so on up to the $k$-th element, for a total of $n^k$.
(ii) There are $n$ choices for the first element, and for every such choice there are $n-1$ choices for the second element. For every choice of elements $a_1$ and $a_2$, there are $n-2$ choices for the third element $a_3$, and so on, for a total of $n(n-1)(n-2)\cdots (n-k+1)$. This can be rewritten in various ways, for example as $\frac{n!}{(n-k)!}$. Your book may also use a "permutation" symbol.
(iii) I don't know what "contains only one ordered pair" means.
(iv) We can choose the $k$ elements in $\binom{n}{k}$ ways. For any such choice, there is only one way to line up the chosen elements in increasing order.
(v) I don't know what tools you have. We are counting the number of $k$-element multisets. Or else, equivalently, we are counting the number of solutions of $x_1+\cdots+x_n=k$, where $x_1$ is the number of $1$'s we choose, $x_2$ is the number of $2$'s, and so on. The standard way to solve this kind of problem is *Stars and Bars (please see Wikipedia). I would prefer not to write down an answer, but if you produce an answer I can tell you whether it is right.
