How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of…
How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?
120
21600
720
3600
Solution
We have the word 'D A U G H T E R'
where vowels $=\mathrm{A}, \mathrm{U}, \mathrm{E}=3_{\mathrm{c}_2}$
consonant $=\mathrm{D}, \mathrm{G}, \mathrm{H}, \mathrm{T}, \mathrm{R}=5_{\mathrm{c}_3}$
$\Rightarrow$ There are $3_{\mathrm{c}_2} \times 5_{\mathrm{c}_3}=30$ ways to select 2 vowels and 3 consonants
$
\Rightarrow 3_{\mathrm{c}_2} \times 5_{\mathrm{c}_3}=30
$
how each letter can be arranged in 5 ! ways
$
\Rightarrow 30 \times 5 !=3600
$