# Thread: Gr12 Permutation Problem with a condition

1. ## Gr12 Permutation Problem with a condition

Hi, I'm just looking for some different ways to solve this.

How many different arrangements of the letters SPECIAL can be made using exactly 2 vowels and exactly 2 consonants?

2. ## Re: Gr12 Permutation Problem with a condition

Hey Pourdo.

Your question is a little bit ambiguous since having only 2 vowels and 2 constants means that you aren't just re-arranging the words but creating a word which has a subset of symbols.

Are you trying to just find the number of ways of creating a new word with two vowels and two consonants?

3. ## Re: Gr12 Permutation Problem with a condition

Hmm sorry if it was confusing. Yes I believe it is asking how many 4 letter arrangements you can create using the letters S P E C I A L but the 4 letter arrangements must have two vowels and two consonants.

Vowels - E I A
Consonants - S P C L

4. ## Re: Gr12 Permutation Problem with a condition

Are you aware of the hyper-geometric distribution?

Hypergeometric distribution - Wikipedia, the free encyclopedia

5. ## Re: Gr12 Permutation Problem with a condition

Hello, Pourdo!

How many different arrangements of the letters SPECIAL can be made
using exactly 2 vowels and exactly 2 consonants?
I would solve it like this . . .

There are 3 vowels:. $\{A,E,I\}$
There are 4 consonants:. $\{C,L,P,S\}$

Select 2 vowels: ${3\choose2} = 3$ ways.
Select 2 consonants: ${4\choose2} = 6$ ways.

Arrange the 4 letters: $4! = 24$ ways.

Therefore:. $3\cdot6\cdot24 \,=\,432$ arrangements.

6. ## Re: Gr12 Permutation Problem with a condition

Thank you! That`s sensible.