Chapter 13.2, Problem 23AYU

List all the ordered arrangements of 5 objects $a$, $b$, $c$, $d$, and $e$ choosing 3 at a time without repetition. What is $P\left(5,3\right)$?

To determine

To write:

a. All the ordered arrangements of 5 objects $a$, $b$, $c$, $d$, and $e$ choosing 3 at a time without repetition.

b. What is $P\left(5, 3\right)?$

Answer to Problem 23AYU

a. $\{abc, abd, abe,acd, acb, ace, adb,adc,ade, aeb, aec, aed, bac,bad, bae, bca, bcd,bce, bda, bdc,bde,bea, bec,bed, cba, cbd, cbe,cad, cab, cae, cdb, cda,cde, ceb, cea, ced, ebc, eba, ebd, eca, ecb, ecd, eab, eac, ead, edb, edc, eda\}$

b. 60

Explanation of Solution

Given:

The 5 objects $a$, $b$, $c$, $d$, $e$ and number of objects to be chosen is 3.

Formula used:

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

Calculation:

The list of ordered arrangements of 5 objects $a$, $b$, $c$, $d$, and $e$ choosing 3 at a time without repetition is as follows

Arrangement starting with the object $a$ is $\left\{abc, abd, abe,acd, acb, ace, adb,adc,ade, aeb, aec, aed\right\}$.

Arrangement starting with the object $b$ is $\{bac,bad, bae, bca, bcd,bce, bda, bdc, bde,bea, bec,bed\}$.

Arrangement starting with the object $c$ is $\left\{cba, cbd, cbe,cad, cab, cae, cdb, cda,cde, ceb, cea, ced\right\}$.

Arrangement starting with the object $d$ is $\left\{ebc, eba, ebd, eca, ecb, ecd, eab, eac, ead, edb, edc, eda\right\}$.

The list of ordered arrangements of 5 objects $a$, $b$, $c$, $d$, and $e$ choosing 3 at a time without repetition is as follows $\{abc, abd, abe,acd, acb, ace, adb,adc,ade, aeb, aec, aed, bac,bad, bae, bca, bcd,bce, bda, bdc,bde,bea, bec,bed, cba, cbd, cbe,cad, cab, cae, cdb, cda,cde, ceb, cea, ced, ebc, eba, ebd, eca, ecb, ecd, eab, eac, ead, edb, edc, eda\}$.

$P\left(5,3\right)=\frac{5!}{\left(5-3\right)!}=\frac{5\text{*}4\text{*}3\text{*}2\text{*}1}{1\text{*}2}=60$