$\mathrm{DAUGHTER}$ શબ્દના મૂળાક્ષરોનો ઉપયોગ કરીને $2$ સ્વરો અને $3$ વ્યંજનો દ્વારા અર્થસભર કે અર્થરહિત કેટલા શબ્દો બનાવી શકાય ?

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 $=30 \times 5 !=3600$