Question

In how many ways can the letters of the word $PERMUTATIONS$ be arranged if the
$\left(i\right)$ Words starts with $P$ and end with $S$.
$\left(ii\right)$ Vowels are all together
$\left(iii\right)$ There are always $4$ letters between $P$ and $S$ ?

Answer 1

$\left(i\right)$ Finding number of words start with $P$ and end with $S$.
Let first position be $P$ and last position be $S$ (both are fixed).
$P................S$
Remaining number of letters other than $P$ and $S=10$
In the word $PERMUTATIONS,2T^{\prime }s$ are repeating. So,
$n=10$, $P_1=2$
$\therefore$ Total number of arrangements $=\frac{n!}{P_1!}=\frac{10!}{2!}$
$=\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1}=1814400$

$\left(ii\right)$ Vowels in word $PERMUTATIONS$ are $E,U,A,I,O$
Using $EUAIO$ as a single letter
Now, vowels are coming together and total letter in $EUAIO=n=5$
Total permutations of $5$ letters $={}^5{P}_5=\frac{5!}{(5-5)!}$
$=\frac{5!}{0!}=\frac{5!}{1}=5\times 4\times 3\times 2\times 1=120$

Arranging remaining letters
Numbers we need to arrange $=7+1=8$
$\therefore n=8$ and $2T^{\prime }s$ are repeating.
So, $P_1=2$
Total arrangement $=\frac{n!}{P_1!}=\frac{8!}{2!}$
$\therefore$ Total number of arrangements $=\frac{8!}{2!}\times 120=2419200$

$\left(iii\right)$ Finding total number of cases.

Position of $P$	$1^{st}$	$2^{nd}$	$3^{rd}$	$4^{th}$	$5^{th}$	$6^{th}$	$7^{th}$	$8^{th}$ (not possible)
Position of $S$	$6^{th}$	$7^{th}$	$8^{th}$	$9^{th}$	${10}^{th}$	${11}^{th}$	${12}^{th}$	${13}^{th}$(not possible)

$\therefore$ Total number of cases when $P$ is before $S=7$
and Total number of cases when $S$ is before $P=7$
$\therefore$ Total number of cases $=7+7=14$ cases
Finding permutations of letters in $1$ case.
$\therefore P$ and $S$ fixed,
so $n=10,P_1=2$
$\because 2T^{\prime }s$ are repeating
$\therefore$ Number of arrangements $=\frac{10!}{2!}$

Finding total number of arrangements
Total number of Permutations $=14\times \frac{10!}{2!}$
$=14\times \frac{10!}{2\times 1}$
$=7\times 10!$
$=25401600$