# Thread: Probability Probelms & Counting Techniques

1. ## Probability Probelms & Counting Techniques

How many arrangements of the word ANXIOUS are there if all of the vowels must come first?

2. Hello, 16sammy!

How many arrangements of the word ANXIOUS are there
if all of the vowels must come first?
There are 4 vowels: $\displaystyle \{A,I,O,U\}$ and 3 consonants: $\displaystyle \{N,S,X\}$

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: .$\displaystyle (4!)(3!) \:=\:(24)(6) \:=\:\boxed{144 \text{ arrangements}}$

3. 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 $\displaystyle 24 \cdot 6\\=144.$