Question

I have counters of $n$ different colours, red , white, blue,....; In how many ways can I make an arrangement consisting of $r$ counters, supposing that there are atleast $r$ of each different colour?

Answer 1

Since there are ${}^{\prime }{n}^{\prime }$ different counters.
So, the first counter can be selected out of ${}^{\prime }{n}^{\prime }$ different counters in ${}^{\prime }{n}^{\prime }$ ways.
Since repetition is allowed here, last counter can also be chosen in ${}^{\prime }{n}^{\prime }$ ways.
$\therefore$ ${}^{\prime }{r}^{\prime }$ counters can be arranged by
$=n\times n\times .....r\left(times\right)$
$=n^r$