How many arrangements of the word ANXIOUS are there if all of the vowels must come first?
Hello, 16sammy!
There are 4 vowels: $\displaystyle \{A,I,O,U\}$ and 3 consonants: $\displaystyle \{N,S,X\}$How many arrangements of the word ANXIOUS are there
if all of the vowels must come first?
Place the 4 vowels first --- there are: $\displaystyle 4!$ possible arrangements.
Then place the 3 constants --- there are: $\displaystyle 3!$ possible arrangements.
Therefore, there are: .$\displaystyle (4!)(3!) \:=\:(24)(6) \:=\:\boxed{144 \text{ arrangements}} $
Lets put the vovels first $\displaystyle \Rightarrow $ (AIOU)(NXS)
So there are $\displaystyle 4 \cdot 3 \cdot 2 \cdot 1$ different ways of arranging the vovels, and for each vowel arrangement there are $\displaystyle 3 \cdot 2 \cdot 1$ arrangements of the consonants $\displaystyle \Rightarrow \ 4! \cdot 3! $ different arrangements giving $\displaystyle 24 \cdot 6\\=144.$