Question

How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together$?$

Answer 1

Step $1$: Finding number of ways of arranging all vowels together.

Vowels in the word EQUATION are E,U,A,I,O

Number of ways of arranging the vowels together = ${}^5{P}_5$

= $\frac{5!}{(5-5)!}$

$=5\times 4\times 3\times 2\times 1$

$=120$

Step $2$ : Finding number of ways of arranging all consonants together.

Consonants in the word EQUATION are Q,T,N

Number of ways of arranging the consonants together
= ${}^3{P}_3$

= $\frac{3!}{(3-3)!}$

$=3\times 2\times 1$

$=6$

Step $3$: Finding total number of ways. EUAIO QTN

Number of ways of arranging $2$ units $=2!=2$

$\therefore$ Total number of ways$=120\times 6\times 2$

$=1440$