Question

The number of different words that can be made with the letters of the word $PLATE$ so that there are exactly two letters between the two vowels is ____ .

Answer 1

$PLATE$ has $2$ vowels $A$ and $E$

Given condition: There are exactly $2$ letters between two vowels

So lets arrange the vowels first.

$-A---E-$ or $E--A-$

Consider case $\left(i\right)-A--E-$

There are $4$ blacks and $3$ letters remaining.

So, we vcan arrange the $3$ letters in $4$ blacks by selecting $3$ blacks and arrange $3$ blacks.

$\to {}^4{C}_3\times 3/\Rightarrow 4\times 6\Rightarrow 29$

Case $\left(ii\right)-E--A-\Rightarrow$ Similar to Case $(i0$

$\therefore$ no. of ways $\Rightarrow {}^4{C}_3\times 3!\Rightarrow 24$

$\therefore$ Total arrang words satisfying given condition $=24+24=48$