# permutation help

#### gogeta

hi, im currently student in vietnam

i have 2 permutation questions

1. how many words consisitng of 3 vowel and 2 consonants can be formed from the letters of the word "columbians"

2. a presidnet secretry and treasurer are to be named from 5 men and 3 women
how many permutations are there if one man and one women must be picked

heres how much i have done

1) you have 4 vowels and 6 consonants and 10 letters total

you need a 5 letter permutation

so you have 4 x 3 x 2 x 6 x 5

but if you do this, will you always have the order in 3 vowels and 2 consonants? what do i multiply this by to mix it up?

keep in mind my total permutations without limits is 10!/5!

2) so i do the same thing as above, i know 5 men, 3 women and a total of 8 ppl

5 x 3 x 6

but wont the man always be president? again, how do i mix this up??

sorry if my english not very good

#### Plato

MHF Helper
1. how many words consisitng of 3 vowel and 2 consonants can be formed from the letters of the word "columbians"
Since you did not say, we assume the letters cannot be repeated.
$$\displaystyle \dbinom{4}{3}\dbinom{6}{2}(5!)$$
Choose the vowels, choose the two consonants, then arrange the five.

#### gogeta

Since you did not say, we assume the letters cannot be repeated.
$$\displaystyle \dbinom{4}{3}\dbinom{6}{2}(5!)$$
Choose the vowels, choose the two consonants, then arrange the five.

P(10,5) which is if I picked 5 random letters from the 10 in COLUMBIANS

#### Plato

MHF Helper
You need to learn to do calculations:
$$\displaystyle \dbinom{4}{3}\dbinom{6}{2}(5!)=7200$$

#### gogeta

Isn't this a permutation question because the order that you place the letters in matters...?

#### gogeta

im unclears as to why you use combinations instead of permutation

#### Plato

MHF Helper
im unclears as to why you use combinations instead of permutation
You must first choose three vowels $$\displaystyle \dbinom{4}{3}=4$$
Then choose the two consonants $$\displaystyle \dbinom{6}{2}=15$$
Now you have five to arrange.
This is a mixed problem.

#### gogeta

You must first choose three vowels $$\displaystyle \dbinom{4}{3}=4$$
Then choose the two consonants $$\displaystyle \dbinom{6}{2}=15$$
Now you have five to arrange.
This is a mixed problem.
thank you i see now that it doesn't matter which vowels you pick, b/c you mix them up by doing 5!

#### gogeta

how would i be able to do the 2nd question, do i multiply by 3!

but then that would be more than P(18,3)

#### Plato

MHF Helper
how would i be able to do the 2nd question
I will give the answer and expect you to explain it.
$$\displaystyle ^N\mathcal{P}_k$$ is the permutation of N taken k at a time.
ANSWER : $$\displaystyle ^8\mathcal{P}_3~-^5\mathcal{P}_3-~^3\mathcal{P}_3$$.
WHY?