Class 11

Math

Algebra

Permutations and Combinations

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?

we have to separate the consonants and vowels and consider each time the set of the consonants and vowels as a single letter.

now,

here, word is E Q U A T I O N

vowels —> E , U, A , I , O ( there are five vowels in given words )

consonants—> T, Q , N ( there are 3 consonants in given words )

the vowels can be arranged in $5!$ ways

the consonants can be arranged in $3!$ ways

These vowels and consonants ( when we take as a single letter )can be arranged $2!$ ways .

hence, a/c to fundamental principle of counting

total number of ways $=5!×3!×2!$

$=(5×4×3×2)×(3×2)×(2)$

$=120×6×2$

$=1440$