Let $S\left(n,k\right)$ denote the number of ways in which $n$ distinct balls can be put into $k$ identical boxes such that no box remains empty $(\mathrm{k}\le \mathrm{n})$. Which of the following is/are correct?

$S(\mathrm{n},\mathrm{k})\equiv$No. of ways to partition n distinct items into k Parts.
First consider one particular element a₁ forming one group and remaining n -1 elements forming k-1 partitions.
Second, consider n - 1elements forming k partitions and a , element going in one of 'k ' groups
Hence, $\mathrm{S}(\mathrm{n},\mathrm{k})=(\mathrm{n}-1,\mathrm{k}-1)+\mathrm{kS}(\mathrm{n}-1,\mathrm{k})$
No. of onto functions = k !S (n, k)
$\mathrm{k}!\mathrm{S}(\mathrm{n},\mathrm{k})=\left[{\mathrm{k}}^{\mathrm{n}}{-}^{\mathrm{k}}{\mathrm{C}}_1(\mathrm{k}-1{)}^{\mathrm{n}}{+}^{\mathrm{k}}{\mathrm{C}}_2(\mathrm{k}-2{)}^{\mathrm{n}}\dots \dots .\right]$
No. of onto functions film 6 elements = 3!S (6,3) to 3 elements