How many permutations can be formed from `10` objects in which `r` objects selected at a time where as two specific objects always occur together?

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)

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 arranging 11 distinct objects taken 4 at a time so that a specified object. (a) always occurs (b) never occurs.

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

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

Find the number of permutations of n different things taken r at a time such that 3 specific things occur together ?

The number of permutations of 'n' different objects taken r at a time, when a particular element always occur in each arrangement, is

n different things are arrnaged around a circle. In how many ways can 3 objects be selected when no two of the selected objects are consecutive?