Help me out with this problem

Question

If the four letter words(need not be meaningful) are to be formed using the letters from the word, "MEDITERRANEAN" such that the first letter is "R" and the fourth letter is "E", then the total number of all such words is?

 

a. 110

b. 59

c. 56

d. (11!)/(2!)^3

Answer 1

There are 13 total letters in the word, including 2 R's and 3 E's.

 

Any possible 4 letter word must read as:

 

R _ _ E

 

So we need to count all of the possibilities for each letter:

 

First letter: 2C1 (either of the R's) = 2

Second letter: 11C1 (any of the remaining letters except the first R and fourth E) = 11
Third Letter: 10C1 (any of the remaining letters except the other three used in the word) = 10

Fourth Letter: 3C1 (any of the E's) = 3

 

All possible combos = 2 * 11 * 10 * 3 = 660

 

 

HOWEVER, this is not asking for all possible combos. It is asking for the total number of unique words.  That means we need to eliminate the repeats. So it does not matter how many combos of R's and E's there are because they are just creating the same words multiple times.  We only need to count these once! So we can change the formula to...  

 

Total unique words = 1 * 11 * 10 * 1 = 110