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)`

Options

A. `""^(n-2)C_(r-2)xxr!`
B. `""^(n-2)C_(r)xxr!xx2!`
C. `""^(n-2)C_(r)" "_(2)xx2!`
D. `""^(n-2)C_(r-2)xx(r-1)!xx2`

Explanation

In order to arrange n distinct items by taking r at a time when two specified objects always occur together we first select (r-2) objects from the remaining (n-2) objects. This can be done in `""^(n-2)C_(r-2)` ways. Now, we consider two specified
objects temporarilly as a single object and mix it with (r-2) selected objest. Now these (r-1) objects can be arranged in (r-1)! ways. But two specified objects can be put together in 2! ways.
Hence, required number of arrangenments is
`""^(n-2)C_(r-2)xx(r-1)!xx2!`