Question

In how many ways can letters of the word ‘ALGEBRA’ be arranged without changing the order of vowels and consonants?

Answer 1

Hint: Here the given question is based on the concept of permutation. Here we have the word ‘ALGEBRA’ where we are not changing the order of the vowels and consonants. Since it is an arrangement, we use permutation concepts and we determine the solution for the question.

Complete step by step solution:
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 ‘ALGEBRA’ we have to find the number of ways where the word is arranged such that the vowels and consonants are arranged without changing its order.
In the word ‘ALGEBRA’ there are 3 vowels namely, A, E, A. and 4 constants.
The number of arrangements of the vowels A, E, A where A is repeated once.
Therefore the arrangements of vowels is \[\dfrac{{3!}}{{2!}}\]
on simplifying we have
\[ \Rightarrow \dfrac{{3 \times 2 \times 1}}{{2 \times 1}}\]
On cancelling the terms which is common in both numerator and denominator and then we have
\[ \Rightarrow 3\,ways\]
Therefore in 3 ways the vowels are arranged in the word ALGEBRA
The number of arrangements of the consonants L, G, B, R, where none of the letters are repeated.
Therefore the arrangements of consonants is \[4!\]
on simplifying we have
\[ \Rightarrow 4 \times 3 \times 2 \times 1\]
On multiplying the terms
\[ \Rightarrow 24\,ways\]
Therefore in 24 ways the consonants are arranged in the word ALGEBRA.
Therefore the number of arrangements in the word ‘ALGEBRA’ without changing the order of vowels and consonants is \[3 \times 24\]
On simplifying it we have
\[ \Rightarrow 72\,ways\]
Therefore in 72 ways the letters of the word ‘ALGEBRA’ are arranged without changing the order of vowels and consonants.
So, the correct answer is “72 ways”.

Note: Here in this question there is no direct method where we apply to get the solution. So first we find the number of ways the letters are arranged in the word. and then we find the number of ways the letters are arranged in the word such that the two A’s are together. Here we must know the permutation concept and the formula \[n! = n \times n - 1 \times ... \times 2 \times 1\]