How many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word $\mathrm{DAUGHTER}$ ?
How many words, with or without meaning, each of $2$ vowels and $3$ consonants can be formed from the letters of the word $\mathrm{DAUGHTER}$ ?
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 of $3$ vowels $=\,^{3} C_{2}=3$
Number of ways of selecting $3$ consonants out of $5$ consonants $=\,^{5} C_{3}=10$
Therefore, number of combinations of $2$ vowels and $3$ consonants $=3 \times 10=30$
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 $=30 \times 5 !=3600$
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