How Many Different Four-Letter Words can be Formed GMAT Problem Solving

Question: 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?

1. 59
2. 11! / (2!*2!*2!)
3. 56
4. 23
5. 11! / (3!*2!*2!*2!)

Answer: A
Solution and Explanation:
Approach Solution 1:

We are left with the following 11 letters: {M, D, I, T, R, EE, AA, NN} out of which 8 are distinct: {M, D, I, T, R, E, A, N.}

We should consider two cases:

1. If the two middle letters are the same, we'd have 3 words: EEER, EAAR, and ENNR.
2. When the sequence of the selection counts, we are essentially choosing 2 letters out of 8 if the two middle letters are distinct, so it's 8P2 = 56.

Total = 56 + 3 = 59.

A is the correct answer.

Approach Solution 2:

After the letters E and R took their positions, we have 11 letters. However, E, A, and N all appear twice. Thus, there are 8 unique letters for every 2 positions.
Eight letters for second place, seven letters for third.

There are 8*7=56 possibilities, but we also need to account for 3 more variants with double letters, EAAR, ENNR, and EEER.
Therefore, the final calculation is 56 + 3 = 59.

A is the correct answer.

Approach Solution 3:

The first step in answering this GMAT permutation problem is: Rearrange two letters you've chosen.
The word "Mediterranean" has 13 letters.
We must construct a 4-letter word with a "E" in the beginning and a "R" at the finish.
We must thus choose two more letters from the remaining 11 letters in addition to E and R.
There are two of each of the remaining five letters—M, D, I, T, and R—as well as two Ns, two E's, and two A's in this group of eleven letters.

The second step in answering this GMAT permutation puzzle is: List all of the potential outcomes.
There are two Ns, two E's, two As, and one of each of the remaining five letters out of the total eleven.
Either two distinct letters, or the same letter, can be found in the second and third positions.

When the two letters are different (Case 1)
Our task is to select two different letters from the eight options.
There are 56 ways to do this (8 x 7).

Case 2: When both letters are identical
There are three options: Ns, Es, or As for the two letters. therefore, three methods.
Total number of possibilities is equal to 59 (56 + 3).

The right choice to make is A.

“How many different four-letter words can be formed (the words don't" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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