Question

List all the ordered arrangements of 4 objects 1, 2, 3, and 4 choosing 3 at a time without repetition. What is $P(4,3)$?

Short answer

The list of ordered arrangements is$123,124,132,134,142,143,213,214,$

$231,234,241,243,312,314,321,324,341,342,412,413,421,423,431,432.$

The value of the given permutation is 24.

Step-by-step solution

Step 1. Given information.

3 objects are chosen from 4 objects 1, 2, 3, and 4 at a time without repetition.

The given permutation is $P(4,3)$.

Step 2. Determine the list of ordered arrangements.

The list of ordered arrangements is:

$$\begin{gathered} 123,124,132,134,142,143, \\ 213,214,231,234,241,243, \\ 312,314,321,324,341,342, \\ 412,413,421,423,431,432 \end{gathered}$$

Step 3. Determine the value of permutation.

We know that,

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

Using the above formula, we get

$\begin{array}{rcl}P(4,3)& =& \frac{4!}{(4-3)!}\\ & =& \frac{4!}{1!}\\ & =& \frac{4\times 3\times 2\times 1}{1}\\ & =& 24\end{array}$