Consider the following statements i . Number of ways of placing ' n '   objects in k bins k ≤ n ) such that no bin is empty is C k - 1 ( n - 1 ) ii . Number of ways of writing a positive integer " n ' into a sum of k positive integers is C k - 1 ( n - 1 ) iii . Number of ways of placing ' n ' objects in k bins such that at least one bin is non-empty is C k - 1 ( n - 1 ) iv . C k n - C k

Step-by-step solution

i . Number of ways of placing ' n '   objects in k bins k ≤ n ) such that no bin is empty is equal to number of integral solution of x 1 + x 2 + x 3 + . . . + x k = n which is = C k - 1 n - 1 ii Let x 1 + x 2 + x 3 + . . . + x k = n Then, number of positive integral solutions are = C k - 1 n - 1 iii . Number of ways of placing ' n ' objects in k bins such that at least one bin is non-empty is equal to the number of non-negative integrals solution of x 1 + x 2 + x 3 + . . . + x k = n which