If $n$ is a positive integer, how many $5-$tuples of integers from $1$ through $n$ can be formed in which the elements of the $5-$tuple are written in increasing order but are not necessarily distinct? In other words, how many $5-$tuples of integers $(h,i,j,k,m)$ are there with $1hijkmn?$

As in Example $9\mathrm{.}6\mathrm{.}3,$ you can represent any ordered $5-$tuple of integers $(h,i,j,k,m$ with $1hijkmn$ as a string of $n$ vertical bars and $5$ crosses, with the position of crosses indicating which

$$

$$ integers from $1$ to $n$ are included in the $5-$tuple. Thus, the number of $5-$tuples is the same as the number of strings of $$ vertical bars and $$ crosses, which is