Question

The number of permutation of n different objects taken r at a time, when p particular objects are always to be included is

Answer 1

P(r, p) × P(n-p, r-p)

Answer 2

To find the number of permutations of $n$ objects taken $r$ at a time including $p$ particular objects:

Method 1: Select $r-p$ objects from the $n-p$ non-particular objects in $\binom{n-p}{r-p}$ ways. Combine these with the $p$ particular objects to get a set of $r$ objects. Arrange these $r$ objects in $r!$ ways. Total = $\binom{n-p}{r-p} \times r!$.

Method 2: Choose $p$ positions out of $r$ for the $p$ particular objects in $\binom{r}{p}$ ways. Arrange the $p$ particular objects in these positions in $p!$ ways ($P(r, p)$ ways). Arrange the remaining $n-p$ objects in the remaining $r-p$ positions in $P(n-p, r-p)$ ways. Total = $P(r, p) \times P(n-p, r-p)$. Both formulas are equivalent.

Answer: The number of permutation is $P(r, p) \times P(n-p, r-p)$.

This can be written as $\frac{r!}{(r-p)!} \times \frac{(n-p)!}{(n-r)!}$.

It can also be written as $r! \times \binom{n-p}{r-p}$.