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The five vowels—$A, E, I, O, U$—along with $15$ $X’s$ are to be arranged in a row such that no $X$ is at an extreme position. Also, between any two vowels, there must be at least $3$ $X’s$. The number of ways in which this can be done is

  1. $1200$
  2. $1800$
  3. $2400$
  4. $3000$
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3 Answers

Best answer
15 votes
15 votes

First, we'll place the vowels

& we can place the $5$ vowels in $5!$ ways

Now, $X$ can't be seated or placed in extreme position.

So,

Now, between any $2$ vowels, there can be at least $3$ X's

So, we're done with $3 \times 4 = 12 $ X's

& $12$ X's can be placed in $4$ ways (as all X's are identicals)

Now, we're left with $(15-12) = 3$ X's to place

Now, these $3$ X's can be placed in many ways-

  • When all the $3$ X's are placed between the same $2$ vowels 

∴ Total ways = $4$

  • Place $2$ X's between the same pair of vowels, & place the remaining X in other pair of vowels

∴ Total ways = $4 \times 3$ $\qquad [∵\text{Once we fill 2 X's, then we'll have only 3 ways to fill 1 X}]$ 

  • Place X's in different pair of vowels

∴ Total ways = $4$

Or, we can compute it in this way, ------

We have $4$ possibilities left & we have to place $3$ X's.

So, $^4C_3 = \dfrac{4!}{3! \times (4-1)!} = \dfrac{4!}{3!} = 4$

$\color{green}{\text{The total no. of ways} = 5! \times \left(4+ 4 \times 3 + 4 \right)}$

$\qquad \qquad = 120 \times \left(4+12+4 \right)$

$\qquad \qquad = 120 \times 20$

$\qquad \qquad = \color{gold}{2400 \hspace{0.1cm} ways}$

Correct Answer: $C$

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18 votes
18 votes
If we consider AxxxExxxIxxxOxxxxxU .we get 4 places here to place x so x1+x2+x3+x4=15 where each x>=3 so it boils down to x1+x2+x3+x4=3 so no of integral solution is 6C3=20 and 5 vowels can be permutated in 5! Ways so total 20×5!=2400
2 votes
2 votes

The first and last place need to have vowels  $\Rightarrow$ 5 * 4 ways

We now have 3 vowels and 18 places remaining.

18/3 = 6 equidistant places for the remaining 3 vowels and therefore $\binom{6}{3}*3!$ ways

totally  5* 4*  $\binom{6}{3}*3!$ = 2400 ways

Answer:

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