Question

If the four letter words (need not to meaningful) are to be formed using the letter 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.\[\dfrac{{11!}}{{{{\left( {2!} \right)}^3}}}\]
D.56

Answer 1

Hint: Here, we will take two cases in consideration. In the first case, we will find out the number of words which can be made when both the selected letters are the same. In the second case we will find out the number of possible words when the 2 selected letters are distinct. Adding both the cases together, we will get the required total number of possible words.

Formula Used:
\[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] , where \[n\] is the total number of letters in the word and \[r\] is the number of letters where vowels can be placed.

Complete step-by-step answer:
According to the question,
The given word is: “MEDITERRANEAN”
It has 13 letters and we are required to form a four letter word such that the first letter is R and the fourth letter is E respectively.
Now, when these 2 letters i.e. R and E are taken out from the 13 letters then, we are left with the following 11 letters:
{M, D, I, T, R, AA, NN, EE}
Clearly, it is having 2 As, 2 Ns and 2 Es , which are the same.
And, we have 8 distinct letters, i.e. {M, D, I, T, R, A, N, E}
Now, we have to select 2 letters from these 11 letters which could be placed in the remaining two digits of the word to be formed.
Now, in order to select those two letters, we will consider two cases:
If the middle two letters are the same, i.e. both of them are either EE or AA or NN respectively.
Clearly, since the letters are the same, hence, they cannot be arranged among themselves. Therefore, we can have only 3 words from this case.
If the middle two words are distinct or different. This means that we are required to choose from the distinct 8 letters, i.e. {M, D, I, T, R, A, N, E}
Now, we have to choose 2 letters from these distinct 8 letters, hence, we will use the formula of combinations.
Therefore, substituting \[n = 8\] and \[r = 2\] in the formula \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\], we get,
\[{}^8{C_2} = \dfrac{{8!}}{{2!\left( {8 - 2} \right)!}}\]
Subtracting the terms in the denominator, we get
\[ \Rightarrow {}^8{C_2} = \dfrac{{8!}}{{2!6!}}\]
Computing the factorial in the numerator, we get
\[ \Rightarrow {}^8{C_2} = \dfrac{{8 \times 7 \times 6!}}{{2!6!}}\]
\[ \Rightarrow {}^8{C_2} = \dfrac{{8 \times 7}}{{2!}}\]
But, these 2 selected letters can also be arranged among themselves in \[2!\] ways.
Therefore, the required number of ways for this case will be: \[{}^8{C_2} \times 2!\]
 \[{}^8{C_2} \times 2! = \dfrac{{8 \times 7}}{{2!}} \times 2!\]
Simplifying the expression, we get
\[ \Rightarrow {}^8{C_2} \times 2! = 8 \times 7 = 56\]
Therefore, the total number of possible cases by adding both the cases together.
Total number of possible words \[ = 3 + 56 = 59\]
Hence, if the four letter words (need not to be meaningful) are to be formed using the letter 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 59.
Therefore, option B is the correct answer.

Note: While solving this question, we should know the difference between permutations and combinations. Permutation is an act of arranging the numbers, whereas combination is a method of selecting a group of numbers or elements in any order. Also, in order to answer this question, we should know that when we compute a factorial then, we write it in the form of: \[n! = n \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times ...... \times 3 \times 2 \times 1\]. By factorial we mean that it is a product of all the positive integers which are less than or equal to the given number but not less than 1.