How many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word DAUGHTER?
Let me tell my combinations approach which got me the right answer:
The given word has $3$ vowels and $5$ consonants.
Number of possible combinations = $^2C_3 \times ^3C_5$. These combinations can be arranged among themselves in $5!$ ways. Hence, number of words = $5 ! \times ^2C_3 \times ^3C_5 = 3600$
Attempt using Permutations:
There are $5$ vacant places. Thus, number of permutations possible = $2!\times 3! \times^3P_2 \times ^5P_3 $. This doesn't give the right answer.
Please tell me how to approach this problem using Permutations only.