Permutations of distinct objects

Prove that number of permutations of n distinct objects taken r at a time; when a particular onject is never taken in each a arrrangement is (n-1)P_(r)

Distribution of n distinct objects into r distinct boxes if in each box at least one box is placed

The no.of all permutation of n distinct things taken all at a time is n!

The number of permutations of n different objects taken r(>=3) at a time which include 3 particular objects is

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

The number of permutation of n different objects, taken r at a line, when repetitions are allowed, is …….. .

Distribution of n distinct object in r different boxes (empty box allowed)

Distribution of n distinct object in r different boxes (empty box not allowed)

Prove that the no.of all permutations of n different objects taken r at a time when a particular object is to be always included in each arrangement is r.(n-1)P_(r-1)