Question

The number of words can be formed from the letters of the words MAXIMUM, if two consonants cannot occur together, is
A) \[4!\]
B) \[3!\, \times 4!\]
C) \[7!\]
D) None of these

Answer 1

Hint: Here the given question is based on the concept of permutation. Here we have the word ‘MAXIMUM’ where we arrange all the vowels in such a way that no two consonants are together. Since it is an arrangement, we use permutation concepts and we determine the solution for the question.

Complete step by step answer:
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
Now consider the given question, here we have the word ‘MAXIMUM’ we have to find the number of ways where the word is arranged such that the two consonants cannot occur together.
In the word ‘MAXIMUM’ there are 2 consonants namely M and X and 3 vowels.
The number of arrangements of the vowels A, I, U, where none of the letters are repeated.
Therefore, the arrangement of Vowels is \[3!\].
The number of arrangements of the consonants M, X, M, M, where M is repeated once.
Therefore, the arrangement of consonants is \[\dfrac{{4!}}{{3!}}\].
Therefore, the number of arrangements in the word ‘MAXIMUM’ without two consonants occur together is
\[ \Rightarrow 3!\, \times \dfrac{{4!}}{{3!}}\]
On simplification, we get
\[\therefore \,\,\,4!\] ways
Hence, in \[4!\] ways the letters of word ‘MAXIMUM’ can be arranged without two consonants occurring together. Therefore, option (A) is the correct answer.

Note:
In Permutation, the number of ways of arranging \[n\] objects in which there are object \[p\], \[q\] and \[r\] identical is given then the ways \[ = \dfrac{{n!}}{{p!\, \cdot \,q!\, \cdot \,r!}}\]. Remember Consonants are all the non-vowel sounds, or their corresponding letters: A, E, I, O, U, there are 21 consonant letters in English alphabets and factorial is the continued product of first ‘\[n\]’ natural numbers is called the “n factorial” and it represented by \[n! = n \cdot \left( {n - 1} \right) \cdot \left( {n - 2} \right) \cdot \left( {n - 3} \right) \cdot .... \cdot 3 \cdot 2 \cdot 1\].