`(a)` The number of ways in which `r` objects can be selected form `n` different objects if `k` particular objects are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (i)` Always included `=text()^( n-k)C_(r-k).`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii)` Never included `= text()^(n-k)C_r.`

`(b)` The number of arrangement of `n` distinct objects taken `r` at a time so that k particular objects are

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (i)` Always included `= text()^(n-k)C_(r-k).r!,`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii)` Never included `= text()^(n-k)C_r.r!.`

`text(Number of Permutations Under Certain Conditions)`

(i) Number of permutations of n different things, taken r at a time, when a particular thing is to be always included in each arrangement, is

`r. text()^(n-1) P_(r-1)`

`text(Corollary)` Number of permutations of n different things, taken r at a time, when p particular things is to be always included in each arrangement, is `p!(r-(p-1)^(n-p))P_(r-p)`

(ii) Number of permutations of n different things, taken r at a time, when a particular thing is never taken in each arrangement, is
`text( )^(n-1)P_r`

(iii) Number of permutations of n different things, taken all at a time, when m specified things always come together, is
`m! xx (n - m + 1) !`

(iv) Number of permutations of n different things, taken all at a time, when m specified things never come together, is
`n! - m! xx (n -m + 1)!`