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

The word DAUGHTER has 3 vowels A, E, U and 5 consonants D, G, H, T and R.

The three vowels can be chosen in

    \[^{3}{{C}_{2}}\]

 as only two vowels are to be chosen.

Similarly, the five consonants can be chosen in

    \[^{5}{{C}_{3}}\]

 ways.

∴ Number of choosing 2 vowels and 5 consonants would be

    \[^{3}{{C}_{2}}~\times {{~}^{5}}{{C}_{3}}\]

NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations Image 38

    \[=\text{ }30\]

∴ Total number of ways of is 30

Each of these 5 letters can be arranged in 5 ways to form different words

    \[\Rightarrow {{~}^{5}}{{P}_{5}}=\]

NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations Image 39

Total number of words formed would be

    \[=\text{ }30\text{ }\times \text{ }120\text{ }=\text{ }3600\]