Find the number of distinct 4 letter words (permutation)

[Guest]

Find the number of distinct 4 letter words containing at least one vowel that can be formed from FERMAT.

all ive done is :

E_ _ _ _
_ E _ _
_ _E_
_ _ _ E

A_ _ _ _
_ A _ _
_ _A_
_ _ _ A

but I dont know how to carry on, thanks

 

[pka]

That is good start. But there is a danger of an over-count in that approach.
This problem is what is known as a “backdoor” problem: count what we don’t want and remove them from the total.

There are (6)(5)(4)(3) ways to make any four letter string of distinct letters from {F,E,R,M,A,T}. There are (4)(3)(2)(1) ways to do it NO vowels. So how many have at least one.

You see “at least one” is the opposite of “none”. Any problem that asks for “at least one” can be done by the ‘backdoor’ approach.

 

[Guest]

so you would do: (4)(3)(2)(1) ?\

or (6)(5)(4)(3)- (4)(3)(2)(1)? im confused on that a little. I don't know when you're suppose to subtract (the total amount of ways) - (the amount of ways that contain a vowel) or to just do (4)(3)(2)(1) and thats the answer..

 

[pka]

so you would do: (4)(3)(2)(1) ?\

or (6)(5)(4)(3)- (4)(3)(2)(1)? im confused on that a little. I don't know when you're suppose to subtract (the total amount of ways) - (the amount of ways that DO NOT contain a vowel) or to just do (4)(3)(2)(1) and thats the answer..