6.1 Permutations Part 2

Posted on April 23, 2020 by user

(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} {(a)}^{5} P_{2} \\ {(b)}^{7} P_{3} \\ {(c)}^{9} P_{4} \end{array}

Solution:

\begin{array}{l} {(a)}^{5} P_{2} = \frac{5!}{(5 - 2)!} = \frac{5!}{3!} \\ = \frac{5 \times 4 \times 3!}{3!} = 5 \times 4 = 20 \\ {(b)}^{7} P_{3} = \frac{7!}{(7 - 3)!} = \frac{7!}{4!} \\ = \frac{7 \times 6 \times 5 \times 4!}{4!} = 7 \times 6 \times 5 = 210 \\ {(c)}^{9} P_{4} = \frac{9!}{(9 - 4)!} = \frac{9!}{5!} \\ = \frac{9 \times 8 \times 7 \times 6 \times 5!}{5!} = 9 \times 8 \times 7 \times 6 = 3024 \end{array}

Calculator Computation: