# How Many Words, with Or Without Meaning, Each of 2 Vowels and 3 Consonants Can Be Formed from the Letters of the Word Daughter? - Mathematics

How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?

#### Solution 1

In the word DAUGHTER, there are 3 vowels namely, A, U, and E, and 5 consonants namely, D, G, H, T, and R.

Number of ways of selecting 2 vowels out of 3 vowels =""^3C_2 = 3

Number of ways of selecting 3 consonants out of 5 consonants = ""^5C_2 = 3

Therefore, number of combinations of 2 vowels and 3 consonants = 3 × 10 = 30

Each of these 30 combinations of 2 vowels and 3 consonants can be arranged among themselves in 5! ways.

Hence, required number of different words = 30 × 5! = 3600

#### Solution 2

In the word DAUGHTER, there are 3 vowels namely, A, U, and E, and 5 consonants namely, D, G, H, T, and R.

Number of ways of selecting 2 vowels out of 3 vowels =""^3C_2 = 3

Number of ways of selecting 3 consonants out of 5 consonants = ""^5C_2 = 3

Therefore, number of combinations of 2 vowels and 3 consonants = 3 × 10 = 30

Each of these 30 combinations of 2 vowels and 3 consonants can be arranged among themselves in 5! ways.

Hence, required number of different words = 30 × 5! = 3600

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#### APPEARS IN

NCERT Class 11 Mathematics Textbook
Chapter 7 Permutations and Combinations
Q 1 | Page 156
NCERT Class 11 Mathematics Textbook
Chapter 7 Permutations and Combinations
Q 1 | Page 156