# Gr12 Permutation Problem with a condition

• Mar 19th 2013, 09:10 PM
Pourdo
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?

• Mar 19th 2013, 09:25 PM
chiro
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?
• Mar 19th 2013, 09:59 PM
Pourdo
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
• Mar 20th 2013, 03:37 AM
chiro
Re: Gr12 Permutation Problem with a condition
Are you aware of the hyper-geometric distribution?

Hypergeometric distribution - Wikipedia, the free encyclopedia
• Mar 20th 2013, 05:57 AM
Soroban
Re: Gr12 Permutation Problem with a condition
Hello, Pourdo!

Quote:

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

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

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

Therefore:. $\displaystyle 3\cdot6\cdot24 \,=\,432$ arrangements.
• Mar 20th 2013, 10:47 AM
Pourdo
Re: Gr12 Permutation Problem with a condition
Thank you! That`s sensible.