How many different 5 letter words can be formed?

[pgfreak]

Sorry for the multiple questions, trying to study for a test)

How many different five letter "words" (dont have to make sense) can be formed from the letters in the word 'stevin' if:
a. only one vowel has to be used
b. both vowels have to be used, and e is to precede the i

I did A. I did 4! x 2! = 48 ... (2) (4) (3) (2) (1)

I cant get B...so I drew it so that I and E and first up and it looks like this (1) (1) (4) (3) (2)
and there is 4 ways to place the i
3 ways to place the e
and I dont know what to do for the other letters to finish off this problem now... I have that there is 4P3 ways for the other letters but I dont think that is correct (I dont think 4 x 3 x 4P3 is correct)
What do I do about the remaining letters?

 

[galactus]

Part b.

Let's see if I count this up correctly. Let's put the 'e' in the first position:

e_ _ _ _ _, arrange the other 5 letters in 5!=120 ways. The 'i' is going to be after the 'e' no matter what.

Put the 'e' in the second place: _ e_ _ _ _, choose 1 of the other 4 letters but the 'i' and put it first, then arrange the other 4, including the 'i'. That's C(4,1)*24=96 ways.

Put the 'e' in the 3rd place: _ _ e_ _ _, choose 2 of the 4 letters but the 'i' and put in the first two places, then arrange the other 3, including the 'i'. C(4,2)*6=36.

Put the 'e' in the fourth place: _ _ _ e _ _, Choose 3 of the 4 letters but the 'i' to put in the first three spots, then arrange the last 2. C(4,3)*2=8

Now, finally, put the e in the fifth place and the 'i' has to go in the final position. _ _ _ _ e i

Arrange the other 4 in 24 ways.

120+96+36+8+24=284

This seems correct, but it's easy to miscount. Check it out.