MEDITERRANEAN combinatorics

[lakita7]

I came across the following problem and struggle to understand how the answer (59) was achieved:

How many different four letter words can be formed (the words need not be meaningful) using the letters of the word "MEDITERRANEAN" such that the first letter is E and the last letter is R?

I am trying to use the following slot method:

1) There are 3 choices for E in the first position, 2 choices for R in the last position, and 11 remaining letters, giving us:
3 x 2 x 11!

2) of the remaining letters, we have 2 E, 2A and 2N where "order doesn't matter", giving us:
(3 x 2 x 11!) / (2! x 2! x 2!)

Which doesn't yield 59... Could you please help me get to the right result utilizing this approach?

[george.kourdin]

think the original idea is correct (start with e and r and figure out the middle)

my thinking was that we want different four letter words so the fact that we have 3 choices for E and 2 choices for R is meaningless because a word E1XXR1 is the same as E2XXR2 (where XX=XX) so we don;t want to multiply 2x3. we just want to find the total amount of combinations that could be formed to fill XX

a) XX is actually XY and the two letters are different, having 8 possible letters to choose from we get 56 combos of XY
b) XX is the same letter. there are 3 possible ways this can be done since 3 Ns, 3 A or 3 Es. notice that we dont have 2Rs because we are already using an R...and voila 3+56 =59

[jnelson0612]

Will you please post the original source for this question?

[lakita7]

Many thanks George! After spending so much time on it I just realized they are indeed asking for 4 letter words... Unbelievable that my brain somehow omitted that. Says a lot about reading the question well!

Here is the original source:
http://questionbank.4gmat.com/mba_prep_ ... obability/

[jnelson0612]

Great, thank you!