Question

List all the ordered arrangements of 5 objects $a,b,c,d$, and $e$choosing 2 at a time without repetition. What is $P(5,2)$?

Short answer

The list of ordered arrangements is:

$ab,ac,ad,ae,ba,bc,bd,be,ca,cb,cd,ce,da,db,dc,de,ea,eb,ec,ed.$

The value of the given permutation is 20.

Step-by-step solution

Step 1. Given information.

2 objects are chosen from 5 objects $a,b,c,d$, and $e$at a time without repetition.

The given permutation is $P(5,2)$.

Step 2. Determine the list of ordered arrangements.

The list of ordered arrangements is:

$$\begin{gathered} ab,ac,ad,ae, \\ ba,bc,bd,be, \\ ca,cb,cd,ce, \\ da,db,dc,de, \\ ea,eb,ec,ed \end{gathered}$$

Step 3. Determine the value of permutation.

We know that,

$P(n,r)=\frac{n!}{(n-r)!}$

Using the above formula, we get

$\begin{array}{rcl}P(5,2)& =& \frac{5!}{(5-2)!}\\ P(5,2)& =& \frac{5!}{3!}\\ & =& \frac{5\times 4\times 3!}{3!}\\ & =& 20\end{array}$