Question

Consider the word $Z=\text{FREEWHEEL}.$ Then which of the following statements is (are) CORRECT?

• A. Number of other different ways in which the letters of the word $Z$ can be arranged is ${}^9{P}_5.$
• B. Number of ways in which the letters of the word $Z$ can be arranged in a circle is ${}^8{P}_4,$ distinguishing between clockwise and anti-clockwise.
• C. Number of ways in which the letters of the word $Z$ can be arranged if vowels and consonants both are in alphabetical order is $\frac{1}{2}\times {}^{10}{C}_5$
• D. If as many more words are formed as possible using the letters of the word $Z,$ the number of words which contain the word $\text{FEEL}$ is ${}^6{P}_4.$

Answer 1

Answer: (B), (C), (D)

The correct options are
B Number of ways in which the letters of the word $Z$ can be arranged in a circle is ${}^8{P}_4,$ distinguishing between clockwise and anti-clockwise.
C Number of ways in which the letters of the word $Z$ can be arranged if vowels and consonants both are in alphabetical order is $\frac{1}{2}\times {}^{10}{C}_5$
D If as many more words are formed as possible using the letters of the word $Z,$ the number of words which contain the word $\text{FEEL}$ is ${}^6{P}_4.$

$Z=\text{F R E E W H E E L}$
Letters can be arranged in $\frac{9!}{4!}={}^9{P}_5$
Excluding $\text{F R E E W H E E L},$ we have ${}^9{P}_5-1$ different ways.


There are $9$ letters out of which $4$ are identical.
$\therefore$ Number of circular arrangement $=\frac{(9-1)!}{4!}={}^8{P}_4$

Vowels are $EEEE$.
$4$ places can be selected in ${}^9{C}_4$ ways. Consonants can be alphabetically arranged only in one way.
$\therefore$ Number of ways $=1\times {}^9{C}_4=126=\frac{1}{2}\cdot {}^{10}{C}_5$

Consider $\text{FEEL}$ as one.
Then, number of arrangement $=\frac{6!}{2!}={}^6{P}_4$