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)`

Find the number of permutations of n different things taken r at a time such that two specific things occur together?

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

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

Find the number of permutations of n things taken r at a time in which two particular things : (i) always occur (ii) never occur.

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

Prove that the number of combinations of n things taken r at a time in which p particular things always occur is .^(n-p)C_(r-p)