Question

The number of ways in which the letter of the word “ARRANGE” can be arranged such that both R do not come together is?
(1) \[360\]
(2) \[900\]
(3) \[1260\]
(4) \[1620\]

Answer 1

Hint: First we will find the total number of ways of arranging the letters of the word ‘ARRANGE’. Then we will find the total number of ways of arranging the letters of the word when both the ‘R’ are together. After subtracting both the obtained values we will get the total number of ways of arranging the letter of the word ‘ARRANGE’ when ‘R’ are not together.

Complete step-by-step answer:
This is the question of permutation and combination so this question will be solved with the help of permutation and combination. Now before solving this question first we should know about what is a permutation and what is a combination.
So permutation is the method of arranging several things in several possible ways and the order of items will be considered while doing the arrangement.
The formula of permutation is given by –
\[{}^{n}{{C}_{r}}=\dfrac{\left| \!{\underline {\,
  n \,}} \right. }{\left| \!{\underline {\,
  n-r \,}} \right. }\]
Where ‘n’ represents the total number of objects from which we have to choose and ‘r’ represents the number of objects that we have to choose from ‘n’.
The combination is the method of arranging several things in several possible ways and the order of items will not be considered while doing the arrangement.
The formula of combination is given by-
\[{}^{n}{{C}_{r}}=\dfrac{\left| \!{\underline {\,
  n \,}} \right. }{\left| \!{\underline {\,
  r \,}} \right. \left| \!{\underline {\,
  n-r \,}} \right. }\]
So in the above question, we have the word ‘ARRANGE’ and we have to find out the total number of ways in which these words can be arranged when the repetition of the letter ‘R’ is not allowed.
First, we will find out the total number of ways in which these letters of the word can be arranged.
So in the word ‘ARRANGE’, the total number of ways to arrange the letter when the repetition is allowed is given by-
\[Total\text{ }no.\text{ }of\text{ }ways=\dfrac{\left| \!{\underline {\,
  7 \,}} \right. }{\left| \!{\underline {\,
  2 \,}} \right. \left| \!{\underline {\,
  2 \,}} \right. }\]……….eq(1)
In the eq(1), we have written the factorial of \[7\]in the numerator because there are total \[7\]letters in the word and we have added two times the factorial of \[2\]in the denominator because the letter ‘A’ and the letter ‘R’ are occurring in the word two times.
So from eq(1)
\[Total\text{ }no.\text{ }of\text{ }ways=\dfrac{7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 2}\]
So the total number of ways of arranging the word ‘ARRANGE’ will come out to be \[1260\].
Now we will find out the total number of ways of arranging the letters of the word when both the letters ‘R’ are allowed to come together.
So we will consider both ‘R’ in the word as one letter and then we are left with six letters in the word. Now, the number of ways of arranging those letters will be given below.
\[Total\text{ }no.\text{ }of\text{ }ways=\dfrac{\left| \!{\underline {\,
  6 \,}} \right. }{\left| \!{\underline {\,
  2 \,}} \right. }\]
We have kept the factorial of \[2\]in the denominator because the letter ‘A’ is still occurring two times in the word. So the result will be as shown below.
\[Total\text{ }no.\text{ }of\text{ }ways=\dfrac{6\times 5\times 4\times 3\times 2\times 1}{2}\]
So the total number of ways of arranging the word when both the ‘R’ are together is \[360\].
Now we have to find the total number of ways of arranging the word when the ‘R’ letter is not together which will be obtained when we subtract the number of ways when ‘R’ are together from the total number of ways of arranging the letters of the word ‘ARRANGE’.
Total no. of ways when ‘R’ is not together= \[1260-360\]
So the total number of ways of arranging the letters of the word ‘ARRANGE’ when ‘R’ are not together are \[900\] ways.
The correct answer will be an option (2)
So, the correct answer is “Option 2”.

Note: Permutation is considered as an ordered combination where the order of arrangement matters but the combination is considered as an unordered combination where the order and selection of the arrangement do not matter. To calculate combination and permutation, you should know how to calculate the factorial.