Number of Combinations of n different objects taken r at a time|Questions

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

The number of permulations of n different objects taken K at a time , when repetitions are allowed is

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

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)

Prove that the no.of permutations of n different objects taken r at a time in which two specified objects always occur together 2!(r-1)(n-2)P_(r-2)

theorem: The no.of all combinations of n distinct objects taken r at a time is given by nC_(-)r=n!/(n-r)!.r! )

What is the total number of combination of n different things taken 1, 2, 3,...,n at a time ?