Question

The maximum number of different permutations of 4 letters of the word “EARTHQUAKE” is-
A.2910
B.2550
C.2190
D.2091

Answer 1

Hint: Here, we have to find the maximum number of different permutations of the word ‘EARTHQUAKE’. First, we will find the number of 4 letter words which can be formed from the given word. Then we will find the number of ways when all the four letters are different, two letters are same and two letters are different and two letters are same and the other two letters are same. We will add all the possible ways in all the cases to find the maximum number of permutations.

Formula Used:
Combination is given by the formula \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\].

Complete step-by-step answer:
We have to find the maximum number of different permutations of 4 letters of the word “EARTHQUAKE”.
Letters in the word is E, A, R, T, H, Q, U, K since two letters A, E in the word “EARTHQUAKE” are repeated so that those letters can be considered only once.
Total number of letters in the word EARTHQUAKE\[ = 8\]
We are considering first when all the four letters are different
Combination is given by the formula \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\].
So by using the combination formula, we get
Number of letters formed when all the four letters are different \[{}^8{C_4} = \dfrac{{8!}}{{\left( {8 - 4} \right)!4!}} \times 4!\]
\[ \Rightarrow \] Number of letters formed when all the four letters are different \[{}^8{C_4} = \dfrac{{8!}}{{4!4!}} \times 4!\]
Multiplying the terms, we get
\[ \Rightarrow \] Number of letters formed when all the four letters are different \[{}^8{C_4} = \dfrac{{8!}}{{4!}}\]
Dividing the factorial, we get
\[ \Rightarrow \] Number of letters formed when all the four letters are different \[{}^8{C_4} = 8 \times 7 \times 6 \times 5\]
Multiplying the terms, we get
\[ \Rightarrow \] Number of letters formed when all the four letters are different \[{}^8{C_4} = 1680\]
Second we are considering when two letters are same and two letters are different.
So by using the combination formula, we get
 Number of letters formed in this condition\[ = \dfrac{{{}^2{C_1} \times {}^7{C_2} \times 4!}}{{2!}} = \dfrac{{2!7!}}{{2!\left( {2 - 1} \right)!1! \times \left( {7 - 2} \right)!2!}} \times 4!\]
Simplifying the expression, we get
\[ \Rightarrow \] Number of letters formed in this condition \[ = \dfrac{{2!7!}}{{2! \times 1!1! \times 5!2!}} \times 4!\]
\[ \Rightarrow \] Number of letters formed in this condition \[ = \dfrac{{7!}}{{2! \times 5!}} \times 4!\]
Computing the factorial, we get
\[ \Rightarrow \] Number of letters formed in this condition \[ = \dfrac{{7 \times 6 \times 4 \times 3 \times 2 \times 1}}{2}\]
Simplifying the expression, we get
\[ \Rightarrow \] Number of letters formed in this condition \[ = 504\]
Third, we are considering when two letters are the same and other two letters are the same.
So by using the combination formula \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\], we get
Number of letters formed in this condition\[ = \dfrac{{4!}}{{2!2!}} = \dfrac{{4!}}{{2 \times 2}}\]
Computing the factorial, we get
\[ \Rightarrow \] Number of letters formed in this condition \[ = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 2}}\]
Simplifying the expression, we get
\[ \Rightarrow \] Number of letters formed in this condition \[ = 2 \times 3 = 6\]
Now we will add all the three cases to get the maximum number of permutations. Therefore, we get
Maximum Number of Permutation\[ = 1680 + 504 + 6\]
Adding the terms, we get
\[ \Rightarrow \] Maximum Number of Permutation\[ = 2190\]
Therefore, the maximum number of different permutations of 4 letters of the word “EARTHQUAKE” is 2190.
Hence, option C is the correct answer.

Note: We know that there is not much difference between permutation and combination. Permutation is the way of arranging numbers in some order whereas combination is the way of selecting items where order doesn’t matter. Both the word combination and permutation is the way of arrangement. We might make a mistake by using the permutation formula instead of the combination formula because the word permutation occurs in the question. This should be avoided because the order of letters is not necessary.