Question

In how many of the permutations of $n$ things taken $r$ at a time will $5$ things (i) always occur, (ii) never occur?

Answer 1

(i) Permutations of n different things taken r at a time, when one particular thing always occurs $=r{.}^{n-1}Pr-1$

(ii) Permutation of n different things taken r at a time when a particular thing never occurs ${=}^{n-1}{P}_r$