Question

In how many ways can the letters of the word "STRANGE' be arranged so that

$\left(i\right)$ The Vowel may appear in the Odd places

$\left(ii\right)$ The Vowels are never separated.

$\left(iii\right)$ The Vowels never come together.

Answer 1

Consider the problem

$\left(i\right)$

There are $7$ letters in the word $STRANGE$, of which $2$ are vowels. and there are four Odd places $(First,third,fifthandseventh)$ where the two vowels are to be placed.And two vowels may be placed in these $4$ places in ${}^4{P}_2=4\times 3=12$ ways. And corresponding to each of these $12$ ways the other $5$ letters may be placed in the $5$ places in $5!$ ways $=5\times 4\times 3\times 2=120$ ways.

So the total number of required arrangement

$\begin{array}{c}=12\times 120\\ =1440ways\end{array}$

$\left(ii\right)$

There are $7$ letters. Since the vowels are not to be separated, so we may regard them as forming one letter.

Thus, there are six letters $S,T,R,N,G\left(AE\right)$. They can be arranged among them self in $6!$ ways and two vowels can again be arranged in $2!$ ways.

So, The total number of arrangement (when vowels are vowels come separated)

$=6!\times 2!=1440ways$

$\left(iii\right)$

The number of arrangements in which the vowels do not come together can be obtained by subtracting from the total number of arrangements in which the vowels come together.

Now, the total number of arrangements is $7!$

And the number of arrangements in which the vowels do not come together

$=7!-6!\times 2!=5040$

Therefore, $5040-1440=3600$