I am in trouble with this question: Given the set $S$={$a,b,c,d,e,f$} and 10 position to arrange these elements. How many combination with repetitions exist if the only allowed must contain at least one of each elements of $S$?
For example a valid configuration can be {$a,a,b,b,c,d,e,f,f,f$} and an incorrect one can be {$a,a,a,a,a,b,b,b,b,b$} .
I try to find a solution that can be applicable to a Set of size $m$, with $n$ position to arrange the elements and with $n$>$m$. My starting assumption is that if I consider the Diophantine equation each unknown of solving equation (fulfilling the request) must have at least $1$ like result. However this does not help me to calculate the number of equations that fulfill the initial question. There is an easy or a tricky way to solve this problem?
You must have heard of a counting technique called stars and bars which gives that the number of ways of placing $n$ identical objects into $k$ distinct bins = $\binom{n+k-1}{k-1}$
Possibly what is bothering you is that here the "objects" are distinct numerals. Then just imagine that the bins are numbered $1$ through $6$ and unnumbered identical objects magically acquire the numeral of the bin they are put in, then it fits the stars and bars format.
Firstly put $1$ object in each bin, $4$ are left to be put, and by stars and bars, # of ways = $\binom{4+6-1}{6-1}$