Question

Different words are being formed by arranging the letters of the word$ARRANGE$. All the words obtained are written in the form of a dictionary
The number of ways in which the consonants occur in alphabetic order is
$\left( a \right){\text{ 360}}$
$\left( b \right){\text{ 105}}$
$\left( c \right){\text{ 240}}$
$\left( d \right){\text{ None of these}}$

Answer 1

Hint: Consonants in the word $ARRANGE$ in alphabetical order will be $GNRR$. And also since there are $5$ gaps where we need to place $AAE$. That is $3$ objects which can be done in how many ways. So this will be calculated and then this can be arranged in $3$ ways hence we can find the total number of ways.

Complete step-by-step answer:
So we have to find out the number of ways in which the consonants will occur in alphabetic order.
So we have the word $ARRANGE$
Since the consonants in this word are $GNRR$ and we can’t arrange these words.
So there are a total of $5$ gaps where we need to place the alphabets $AAE$.
And therefore it can be arranged in the ways are-
$ \Rightarrow {}^{3 + 5 + 1}{C_{5 - 1}} = {}^7{C_4} = 35{\text{ ways}}$.
So these may be arranged in several ways are as-
$ \Rightarrow \dfrac{{3!}}{{2!}} = 3{\text{ ways}}$
Thus the total number of ways will be calculated as
$ \Rightarrow 35 \times 3 = 105{\text{ }} ways$
Therefore, the number of ways will be $105$.
Hence, the option $\left( b \right)$ is correct.

Additional Information:
Permutation means all possible arrangements of the number, letter, or any other product or things, etc. Well, the most basic difference in that permutations is ordered sets. The permutation is a course of action of things where the request for a game plan matters. The situation of everything in change matters. Consequently, Permutation can be related to Position.

Note: In Permutation, arrangement matters in Combination, the arrangement does not matter. So while solving this type of problem we should have to keep in mind whether it is of arrangement or it is of grouping. So we should always be kept in mind while solving such types of problems so that we cannot get the error and can solve it easily without any problem.