Consider the following statements regarding $n$ identical objects and $k$ distinct bins:
$i.$ The number of ways of placing $n$ identical objects in $k$ distinct bins $(k \leq n)$ such that no bin is empty is ${}^{n-1}C_{k-1}$.
$ii.$ The number of ways of writing a positive integer $n$ as a sum of $k$ positive integers is ${}^{n-1}C_{k-1}$.
$iii.$ The number of ways of placing $n$ identical objects in $k$ distinct bins such that at least one bin is non-empty is ${}^{n-1}C_{k-1}$.
$iv.$ ${}^nC_k - {}^{n-1}C_k = {}^{n-1}C_{k-1}$.
Which of the above statements are true?

• A
  All the four statements
• B
  $(iii)$ and $(iv)$ only
• C
  All except $(iii)$
• D
  All except $(i)$