Question

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

Answer 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 = \(^3C_2\)=10
Number of ways of selecting 3 consonants out of 5 consonants = \(^5C_3\)=10
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 = \(30 \times 5!\) = 3600