Question

Question 2 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?

Class X1 - Maths -Permutations and Combinations Page 156

Answer 1

first of all, 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

Answer 2

Step-by-step explanation:

$equation \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\: vowels(aeiou) \\ consonants(qtn) \\ \: vowels \: arrangd \: in \: 5factorial \\ con = 3fac \\ both \: arranged \: in \: 2fac \\ so \: total \: \\ 2fac \times 5fac \times 3fac = 1440ways$

hope you understand