# 4.1 Permutations Part 2

(C) Permutation of n Different Objects, Taken r at a Time

1. The number of permutations of n different objects, taken r at a time is given by:

2.  A permutation of n different objects, taken r at a time, is an arrangement of a set of r objects chosen from n objects. The order of the objects in the chosen set is taken into consideration.

3. The number of permutations of n different objects, taken all at a time, is:

Example 1:
Evaluate each of the following.
$\begin{array}{l}{\text{(a)}}^{5}{P}_{2}\\ {\text{(b)}}^{7}{P}_{3}\\ {\text{(c)}}^{9}{P}_{4}\end{array}$

Solution:
$\begin{array}{l}{\text{(a)}}^{5}{P}_{2}=\frac{5!}{\left(5-2\right)!}=\frac{5!}{3!}\\ =\frac{5\text{}×4\text{}×3!}{3!}=5×4=20\\ \\ {\text{(b)}}^{7}{P}_{3}=\frac{7!}{\left(7-3\right)!}=\frac{7!}{4!}\\ =\frac{7\text{}×6\text{}×5\text{}×4!}{4!}=7×6×5=210\\ \\ {\text{(c)}}^{9}{P}_{4}=\frac{9!}{\left(9-4\right)!}=\frac{9!}{5!}\\ =\frac{9\text{}×8\text{}×7\text{}×6\text{}×5!}{5!}=9×8×7×6=3024\end{array}$

Calculator Computation: