In how many ways can be the letters of the word ARRANGE be arranged so that
(i) The two $R^{'} s$ are never together
(ii) The two $A^{'} s$ are together but not two $R^{'} s$
(iii) Neither two $A^{'} s$ nor the two $R^{'} s$ are together.

Question

In how many ways can be the letters of the word ARRANGE be arranged so that
(i) The two

R^{'} s

are never together
(ii) The two

A^{'} s

are together but not two

R^{'} s

(iii) Neither two

A^{'} s

nor the two

R^{'} s

are together.

Filo · Accepted Answer

There are 7 letters 2A′s,2R′s and 3 different.(i) Two R′s and thus we have to arrange remaining 5 letters in which 2A′s are alike and we can have2!5!​=5×4×3=60 ways.After arranging these five letters in a line we have 6 gaps in which 2 different words can be placed in 6P2​ ways. But these two Rs are alike and hence the number of ways will be.2!1​⋅6P2​=2!1​⋅4!6!​=15.Hence the number of ways when Rs are never together is60×15=900, by Fundamental Th.(ii) The two As ate together but npt two Rs.Let us ignore the two Rs and thus we have 7−2=5 letters.Treat the aAs as one letter (AA)NGE thus we have 4 letters which can be arranged in4!2!2!​=24 ways.×AA×N×G×E×between these 4 letter (As together) we have 5 gaps in which 2 different letters can be arranged in 5P2​ways.But here 2Rs are alike and hence the number of ways will be2!1​5P2​=21​⋅3!5!​=10 ways.Hence the number of words when As are together and Rs are not together is 24×10=240 by fundamental theorem.(iii) Neither 2As not 2Rs are together.From part (i), 2Rs are never together in 900 ways.This includes the ways when 2As may be together and may not be together.From part (ii) no. ways that 2As are together but not 2Rs=240.hence the number of ways when neither 2As not 2Rs are together is 900−240=660.

Question Text	In how many ways can be the letters of the word ARRANGE be arranged so that (i) The two $R^{'} s$ are never together (ii) The two $A^{'} s$ are together but not two $R^{'} s$ (iii) Neither two $A^{'} s$ nor the two $R^{'} s$ are together.
Answer Type	Text solution:1
Upvotes	150

In how many ways can be the letters of the word ARRANGE be arranged so that
(i) The two $R^{'} s$ are never together
(ii) The two $A^{'} s$ are together but not two $R^{'} s$
(iii) Neither two $A^{'} s$ nor the two $R^{'} s$ are together.

Text solutionVerified

Practice questions from similar books

In how many ways can be the letters of the word ARRANGE be arranged so that(i) The two R′s are never together(ii) The two A′s are together but not two R′s(iii) Neither two A′s nor the two R′s are together.

Text solutionVerified

Practice questions from similar books

In how many ways can be the letters of the word ARRANGE be arranged so that
(i) The two $R^{'} s$ are never together
(ii) The two $A^{'} s$ are together but not two $R^{'} s$
(iii) Neither two $A^{'} s$ nor the two $R^{'} s$ are together.