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)`

Options

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

Explanation

In order to arrange n distinct items by taking r at a time when a particular object is to be always included in each arrangement, let us first put aside the sepecified item and select (r-1) items from the remaining (n-1) items. This can be done in `""^(n-1)C_(r-1)` ways. Now, we have r items, namely, one specified item and (r-1) selected items. These r items can be arranged in r! ways. Hence, the required number of arrangements is `""^(n-1)C_(r-1)xxr!`