Class 11

Math

Algebra

Permutations and Combinations

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 ____ .

- 48

$PLATE$ 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 $(i)−A−−E−$

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.

$→_{4}C_{3}×3/⇒4×6⇒29$

Case $(ii)−E−−A−⇒$ Similar to Case $(i0$

$∴$ no. of ways $⇒_{4}C_{3}×3!⇒24$

$∴$ Total arrang words satisfying given condition $=24+24=48$